Libro de Algebra

Math 100G/L

Introduction to

Algebra and

Finance

BYU-Idaho

MESSAGE FROM THE FIRST PRESIDENCY

Dear Brothers and Sisters: Latter-day Saints have been counseled for many years to prepare for adversity by having a little money set aside. Doing so adds immeasurably to security and well-being. Every family has a responsibility to provide for its own needs to the extent possible. We encourage you wherever you may live in the world to prepare for adversity by looking to the condition of your finances. We urge you to be modest in your expenditures; discipline yourselves in your purchases to avoid debt. Pay off debt as quickly as you can, and free yourselves from this bondage. Save a little money regularly to gradually build a financial reserve. If you have paid your debts and have a financial reserve, even though it be small, you and your family will feel more secure and enjoy greater peace in your hearts. May the Lord bless you in your family financial effort. The First Presidency

(From the pamphlet ALL IS SAFELY GATHERED IN: FAMILY FINANCES published by the Church.)

Table of Contents

Chapter 1 Arithmetic……….…………………………………………………………...1

Section 1.1………………………………………………………………………………...2 Addition and Multiplication Facts from 1+1 to 15 × 15

Section 1.2……………………………………………………………………………..…10 Rounding and Estimation; Life Plan

Section 1.3………………………………………………………………………………..15 Add, Subtract, Multiply, Divide Decimals; Income and Expense

Section 1.4……………………………………………………………………………..…34 Add, Subtract, Multiply, Divide Fractions; Unit Conversions

Chapter 2 Calculators and Formulas……………………………………………49

Section 2.1…………………………………………………………………………..……50 Exponents Introduction, Order of Operations, Calculator Usage

Section 2.2……………………………………………………..…………………………61 Variables and Formulas

Section 2.3……………………………………………………..…………………………76 Formulas and Spreadsheet Usage

Chapter 3 Algebra.………………....……………………………………………………89

Section 3.1……………………………………………………………………………….90 Linear Equations and Applications

Section 3.2………………………………………………………………………………108 Linear Equations with Fractions; Percent Applications

Section 3.3………………………………………………………………………………121 Exponents Revisited; Loan Payment and Savings Equations

Chapter 4 Graphs and Charts…………………………………..………………...137

Section 4.1………………………………………………………………………………138 Maps and Coordinate Graphs

Section 4.2………………………………………………………………………………148 Graphing Lines and Finding Slope

Section 4.3………………………………………………………………………………162 Using Slope and Writing Equations of Lines

Chapter 1:

ARITHMETIC

Overview

Arithmetic 1.1 Facts 1.2 Rounding and Estimation 1.3 Decimals

1.4 Fractions

1

Section 1.1

2

Section 1.1 Facts Everyone has to start somewhere, and that start, for you, is right here.

When you first started learning math, you probably learned the names for numbers, and then you started to add: 3apples + 7apples equals how many apples? Well 10, of course. My guess is that you caught on to what you were doing and can now add M&M’s, coconuts, gallons of water, money etc. From the beginning I am going to assume you know how to add in your head up to 15+15. If you don’t, please make up some flash cards and get those in your brain. It is similar to learning the alphabet before learning to read. We need the addition facts to be available for instant recall. Soon after addition was learned, I bet someone told you that there was a shortcut when you had to add some numbers over and over. For example: 3+3+3+3+3+3+3 = 21

7 If you notice, there are seven 3’s. 3, seven times, turns out to be 21, so we write it as 7×3 = 21.

One of the best coincidences of the world is that 7, three times, is also 21. 3×7 = 21 Such a switching works for any numbers we pick: 4×5 = 20 and 5×4 = 20

3×13 = 39 and 13×3 = 39

Since we will be using the multiplication facts almost as much as we will be using the addition facts, you need to also memorize the multiplication facts up to 15×15. Learn them well, and you will be able to catch on to everything else quite nicely.

2

Section 1.1 Exercises Part A 1. Make flash cards up to 15+15 and 15×15.

2. Memorize the addition and multiplication facts up to 15+15 and 15×15.

3. Fill out the Addition/Subtraction Monster. Time yourself. Write the time it takes on the

paper. Correct the Addition/Subtraction Monster using your flashcards.

4. Fill out the Multiplication Monster. Time yourself. Write the time on the paper. Correct

the Addition/Subtraction Monster using your flashcards.

Assignment 1.1a

3

Addition/Subtraction Monster Name __________________

12 + 13 = 5 + 6 = 5 + 10 = 12 − 9 = 5 + 9 = 8 + 11 = 5 + 11 = 14 – 4 =

6 + 6 = 7 + 12 = 15 – 8 = 10 + 10 = 10 – 7 = 6 + 11 = 6 + 12 = 6 + 13 =

7 + 7 = 14 – 7 = 7 + 9 = 9 + 13 = 6 + 14 = 15 – 5 = 11 + 11 = 7 – 5 =

12 − 4 = 10 + 12 = 8 + 10 = 13 − 8 = 5 + 5 = 8 + 13 = 5 + 12 = 7 + 8 =

9 + 9 = 5 + 15 = 9 + 11 = 9 + 12 = 15 − 6 = 13 − 5 = 9 + 15 = 8 + 15 =

6 + 7 = 13 − 9 = 8 + 12 = 10 + 13 = 10 + 14 = 10 + 15 7 + 13 = 11 + 13 =

5 + 7 = 11 + 12 = 14 − 9 = 11 + 14 = 11 + 15 = 8 + 9 = 10 − 6 = 8 − 7 =

12 + 12 = 6 + 10 = 12 + 14 = 8 + 8 = 12 − 7 = 12 − 8 = 14 + 14 = 12 − 6 =

9 − 7 = 13 + 14 = 10 − 5 = 7 + 14 = 6 + 9 = 13 − 7 = 13 − 6 = 9 + 10 =

6 + 8 = 14 + 15 = 14 − 10 = 12 + 15 = 14 − 8 = 8 + 14 = 14 − 6 = 10 + 11 =

8 − 5 = 15 − 11 = 15 − 10 = 15 − 9 = 9 − 8 = 7 + 10 = 9 + 14 = 13 + 15 =

7 + 11 = 5 + 14 = 6 + 15 = 15 − 7 = 5 + 13 = 7 + 15 = 5 + 8 = 7 − 6 =

13 + 13 = 8 − 6 = 9 − 5 = 9 − 6 = 15 − 4 = 15 + 15 = 13 − 4 = 14 − 5 =

Time_________

Assignment 1.1 a

4

Multiplication Monster Name __________________

12×13= 5×6= 5×10= 12×9= 5×9= 8×11= 5×11= 14×4=

6×6= 7×12= 15×8= 10×10= 10×7= 6×11= 6×12= 6×13=

7×7= 14×7= 7×9= 9×13= 6×14= 15×5= 11×11= 7×5=

12×4= 10×12= 8×10= 13×8= 5×5= 8×13= 5×12= 7×8=

9×9= 5×15= 9×11= 9×12= 15×6= 13×5= 9×15= 8×15=

6×7= 13×9= 8×12= 10×13= 10×14= 10×15 7×13= 11×13=

5×7= 11×12= 14×9= 11×14= 11×15= 8×9= 10×6= 8×7=

12×12= 6×10= 12×14= 8×8= 12×7= 12×8= 14×14= 12×6=

9×7= 13×14= 10×5= 7×14= 6×9= 13×7= 13×6= 9×10=

6×8= 14×15= 14×10= 12×15= 14×8= 8×14= 14×6= 10×11=

8×5= 15×11= 15×10= 15×9= 9×8= 7×10= 9×14= 13×15=

7×11= 5×14= 6×15= 15×7= 5×13= 7×15= 5×8= 7×6=

13×13= 8×6= 9×5= 9×6= 15×4= 15×15= 13×4= 14×5=

Time_________

Assignment 1.1a

5

Section 1.1 Exercises Part B

Addition/Subtraction Monster 2

9 − 6 = 12 − 4 = 5 + 10 = 6 + 15 = 15 − 5 = 8 + 11 = 12 − 9 = 14 − 4 =

6 + 6 = 9 − 7 = 15 − 8 = 10 + 10 = 10 − 7 = 6 + 11 = 13 − 7 = 5 + 8 =

7 + 7 = 7 + 12 = 15 − 10 = 9 + 13 = 6 + 14 = 12 + 13 = 7 − 5 = 13 + 15 =

5 + 11 = 10 + 12 = 8 + 10 = 15 − 7 = 14 − 7 = 8 + 13 = 5 + 12 = 7 + 8 =

9 + 9 = 5 + 15 = 9 + 11 = 9 + 12 = 6 + 13 = 5 + 5 = 9 + 15 = 8 + 15 =

6 + 7 = 11 + 15 = 8 + 12 = 13 − 5 = 10 + 14 = 10 + 15 = 7 + 13 = 11 + 13 =

5 + 7 = 11 + 12 = 11 + 11 = 11 + 14 = 13 − 8 = 8 + 9 = 10 − 6 = 5 + 9 =

12 + 12 = 14 − 9 = 12 + 14 = 8 + 8 = 12 − 7 = 10 + 13 = 14 + 14 = 12 − 6 =

15 + 15 = 13 + 14 = 10 − 5 = 7 + 14 = 12 − 8 = 6 + 8 = 13 − 6 = 9 + 10 =

5 + 6 = 14 + 15 = 6 + 10 = 12 + 15 = 14 − 8 = 8 + 14 = 14 − 6 = 10 + 11 =

8 − 5 = 15 − 11 = 13 − 9 = 15 − 9 = 6 + 9 = 7 + 10 = 9 + 14 = 7 − 6 =

7 + 11 = 5 + 14 = 15 − 6 = 6 + 12 = 14 − 10 = 7 + 15 = 9 − 8 = 7 + 9 =

13 + 13 = 8 − 6 = 9 − 5 = 5 + 13 = 15 − 4 = 8 − 7 = 13 − 4 = 14 − 5 =

Assignment 1.1b

Time_________

6

Multiplication Monster 2

9×6= 12×4= 5×10= 6×15= 15×5= 8×11= 12×9= 14×4=

6×6= 9×7= 15×8= 10×10= 10×7= 6×11= 13×7= 5×8=

7×7= 7×12= 15×10= 9×13= 6×14= 12×13= 7×5= 13×15=

5×11= 10×12= 8×10= 15×7= 14×7= 8×13= 5×12= 7×8=

9×9= 5×15= 9×11= 9×12= 6×13= 5×5= 9×15= 8×15=

6×7= 11×15= 8×12= 13×5= 10×14= 10×15= 7×13= 11×13=

5×7= 11×12= 11×11= 11×14= 13×8= 8×9= 10×6= 5×9=

12×12= 14×9= 12×14= 8×8= 12×7= 10×13= 14×14= 12×6=

15×15= 13×14= 10×5= 7×14= 12×8= 6×8= 13×6= 9×10=

5×6= 14×15= 6×10= 12×15= 14×8= 8×14= 14×6= 10×11=

8×5= 15×11= 13×9= 15×9= 6×9= 7×10= 9×14= 7×6=

7×11= 5×14= 15×6= 6×12= 14×10= 7×15= 9×8= 7×9=

13×13= 8×6= 9×5= 5×13= 15×4= 8×7= 13×4= 14×5=

Assignment 1.1b

Time_________

7

Section 1.1 Exercises Part C

Addition/Subtraction Monster Name __________________

12 + 13 = 5 + 6 = 5 + 10 = 12 − 9 = 5 + 9 = 8 + 11 = 5 + 11 = 14 − 4 =

6 + 6 = 7 + 12 = 15 − 8 = 10 + 10 = 10 − 7 = 6 + 11 = 6 + 12 = 6 + 13 =

7 + 7 = 14 − 7 = 7 + 9 = 9 + 13 = 6 + 14 = 15 − 5 = 11 + 11 = 7 − 5 =

12 − 4 = 10 + 12 = 8 + 10 = 13 − 8 = 5 + 5 = 8 + 13 = 5 + 12 = 7 + 8 =

9 + 9 = 5 + 15 = 9 + 11 = 9 + 12 = 15 − 6 = 13 − 5 = 9 + 15 = 8 + 15 =

6 + 7 = 13 − 9 = 8 + 12 = 10 + 13 = 10 + 14 = 10 + 15 7 + 13 = 11 + 13 =

5 + 7 = 11 + 12 = 14 − 9 = 11 + 14 = 11 + 15 = 8 + 9 = 10 − 6 = 8 − 7 =

12 + 12 = 6 + 10 = 12 + 14 = 8 + 8 = 12 − 7 = 12 − 8 = 14 + 14 = 12 − 6 =

9 − 7 = 13 + 14 = 10 − 5 = 7 + 14 = 6 + 9 = 13 − 7 = 13 − 6 = 9 + 10 =

6 + 8 = 14 + 15 = 14 − 10 = 12 + 15 = 14 − 8 = 8 + 14 = 14 − 6 = 10 + 11 =

8 − 5 = 15 − 11 = 15 − 10 = 15 − 9 = 9 − 8 = 7 + 10 = 9 + 14 = 13 + 15 =

7 + 11 = 5 + 14 = 6 + 15 = 15 − 7 = 5 + 13 = 7 + 15 = 5 + 8 = 7 − 6 =

13 + 13 = 8 − 6 = 9 − 5 = 9 − 6 = 15 − 4 = 15 + 15 = 13 − 4 = 14 − 5 =

Assignment 1.1 c

Time_________

8

Multiplication Monster Name __________________

12×13= 5×6= 5×10= 12×9= 5×9= 8×11= 5×11= 14×4=

6×6= 7×12= 15×8= 10×10= 10×7= 6×11= 6×12= 6×13=

7×7= 14×7= 7×9= 9×13= 6×14= 15×5= 11×11= 7×5=

12×4= 10×12= 8×10= 13×8= 5×5= 8×13= 5×12= 7×8=

9×9= 5×15= 9×11= 9×12= 15×6= 13×5= 9×15= 8×15=

6×7= 13×9= 8×12= 10×13= 10×14= 10×15 7×13= 11×13=

5×7= 11×12= 14×9= 11×14= 11×15= 8×9= 10×6= 8×7=

12×12= 6×10= 12×14= 8×8= 12×7= 12×8= 14×14= 12×6=

9×7= 13×14= 10×5= 7×14= 6×9= 13×7= 13×6= 9×10=

6×8= 14×15= 14×10= 12×15= 14×8= 8×14= 14×6= 10×11=

8×5= 15×11= 15×10= 15×9= 9×8= 7×10= 9×14= 13×15=

7×11= 5×14= 6×15= 15×7= 5×13= 7×15= 5×8= 7×6=

13×13= 8×6= 9×5= 9×6= 15×4= 15×15= 13×4= 14×5=

Assignment 1.1 c

Time_________

9

Section 1.2Rounding and Estimation

Section 1.2

Now, you know that some arithmetic problems may get long and tedious, so you can understand why some folks choose to estimate and round numbers. Rounding is the quickest, so we will tackle that first. In rounding, we decide to not keep the exact number that

someone gave us. For example:

Rounding If I have $528.37 in the bank, I might easily say that I have about $500. I have just rounded to the nearest hundred. On the other hand, I might be a little more specific and say that I have about (still not exact) $530. I have just rounded to the nearest ten. Here are the places: Just to make sure you are clear on it, here is a big example:

6,731,239,465.726409

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Example: Round to the nearest hundredth: 538.4691 This number is right between 538.46 and 538.47 Which one is nearest? The 9 tells us that we are closer to 538.47 2nd Example: Round to the nearest thousand: 783,299.4321 This number is right between 783,000 and 784,000 Which one is nearest? The 2 in the hundreds tells us that we are closer to : 783,000

10

LAST EXAMPLE

Round $4,278.23 to the nearest hundred 00 Decide if our number is closer to the nearest

hundred above the number or below the number $4,300.$4,278.23 $4,200.00

$4,278.23 ≈ $4,300.00 Change our number to the one it is closer to Answer: $4,300.00

Estimation

Section 1.2

Estimation 1. Round to the highest value. 2. Do the easy problem.

Once rounding is understood, it can be used as a great tool to make sure that we have not missed something major in our computations. If we have a problem like: 3,427,000 × 87.3 We could see about where the answer is if we estimate first: Round each number to the greatest value you can 3,000,000 × 90 Voila! Our answer will be about 270,000,000 We should note that the real answer is: 299,177,100 but the estimation will let us know that we are in the right ball park. It ensures that our answer makes sense.

LAST EXAMPLE Multiply by rounding: 986.7 × 4.9

986.7 ≈ 1,000 Round the numbers

4.9 ≈ 5 1,000 × 5 = 5,000 Multiply the rounded numbers together

986.7 × 4.9 ≈ 5,000 Our answer for 986.7 × 4.9 will be about 5,000

11

Section 1.2 Exercises Part A

1. Round 3,254.07 to the nearest ten.

2. Round 2,892.56 to the nearest tenth.

3. Round 39,454 to the nearest ten thousand.

4. Round 189 to the nearest ten.


6. Round 2,892.56 to the nearest hundred.

7. Round 39, 454 to the nearest ten.

8. Round 189 to the nearest hundred.

Estimate the following.

9. 21 × 3250.07 10. 138.9 × 2892 11. 42 × 189

12. 369.456 ÷ 3.987 13. 58 × 39 14. 351 × 44

Preparation:

15. Find the monthly income for 5 different jobs and be ready to share them with your group.

Answers: 1. 3,250 9. About 60,000 2. 2,892.6 10. About 300,000 3. 40,000 11. About 8,000 4. 190 12. About 100 5. 3,250.1 13. About 2,400 6. 2,900 14. About 16,000 7. 39,450 15. Discuss it together 8. 200

Assignment 1.2a

12



2. Round 2,862.843 to the nearest hundredth.

3. Round 538,484 to the nearest ten thousand.

4. Round 189.59 to the nearest ten.


6. Round 2,892.56385 to the nearest thousandth.

7. Round 34,454 to the nearest thousand.

8. Round 189,364,529.83 to the nearest million.

9. Describe what possible problems students could have with rounding.


10. 51 × 3250.07 11. 438.9 × 2,892.07 12. 32 × 789

13. 569.456 ÷ 6.1987 14. 58 × 391 15. 54,200 ÷ 12

16. Working with your group, find the yearly income for 10 of the jobs brought in by group members. 17. As a group, estimate a monthly budget for a family with a few children living in your area. Please include estimates of costs for housing, transportation, food, utilities, and clothing. 18. Enter the budget into a spreadsheet document.

Answers: 1. 7,250 10. About 150,000 2. 2,862.84 11. About 1,200,000 3. 540,000 12. About 24,000 4. 190 13. About 100 5. 3,250.6 14. About 24,000 6. 2,892.564 15. About 5,000 7. 34,000 16. Make sure they are all there. 8. 189,000,000 17. Should look neat. 9. d vs. dth, lack of 1th, any others 18. Complete when everyone can do it.

Assignment 1.2b

13





4. Round 139.79 to the nearest ten.

5. Round 3,250.647 to the nearest hundredth.

6. Round 2,892.56385 to the nearest thousand.


8. Round 189,364,529.83 to the nearest ten million.


9. 41 × 7250.07 10. 43 × 9.07 11. 82 × 2,890

12. 639.456 ÷ 6.1987 13. 58 × 391.04 14. 56,200 ÷ 12

Begin “Life Plan” Portfolio Project.

15. Imagine your life five years from now. Estimate one month of what you think your expenses and income will be at that time.. 16. Create your own spreadsheet document to record your one month estimated expenses and income. Remember, you are forecasting five years into the future and recording a one month estimate of your anticipated income and expenses into a spreadsheet. Prepare for “Budget and Expenses” Portfolio Project. 17. Report to your group that you have started keeping track of your income and expenses. 18. Receive reports from your group members that they have started tracking their current income and expenses. Answers: 1. 7,254.1 10. About 360 2. 2,860 11. About 240,000 3. 538,000 12. About 100 4. 140 13. About 24,000 5. 3,250.65 14. About 6,000 6. 3,000 15. Include any expenses you can think of. 7. 34,000 16. Save it as “Life Plan”. You will submit it

to your teacher in this lesson. 8. 190,000,000 17. Start your record, then report to your

progress to your group by email, phone, letter, carrier pigeon…

9. About 280,000 18. Complete when everyone has done it.

14

Section 1.3 Decimals

Section 1.3

DEFINITIONS & BASICS

1) Like things – In addition and subtraction we must only deal with like things.

Example: If someone asks you

5 sheep + 2 sheep = you would be able to tell them 7 sheep.

What if they asked you 5 sheep + 2 penguins =

We really can’t add them together, because they aren’t like things.

2) We do not need like things for multiplication and division.

3) Negative – The negative sign means “opposite direction.”

Example: −5.3 is just 5.3 in the opposite direction −5.3 0 5.3 Example : − � is just � in the opposite direction.

� �

Example: −7 – 5 = −12, because they are both headed in that direction

4) Decimal – Deci is a prefix meaning 10. Since every place value is either 10 times

larger or smaller than the place next to it, we call each place a decimal place.

5) Place Values – Every place on the left or right of the decimal holds a certain value

Arithmetic of Decimals, Positives and Negatives

LAWS & PROCESSES

Addition of Decimals

1. Line up decimals

2. Add in columns

3. Carry by 10’s

15

EXAMPLE

Add. 3561.5 + 274.38 3561.5 + 274.38

1. Line up decimals

3 5 6 1. 5 + 2 7 4. 3 8 5. 8 8

2. Add in columns

1

3 5 6 1. 5 + 2 7 4. 3 8 3 8 3 5. 8 8

3. Carry by 10’s. Carry the 1 and leave the 3.

Subtraction of Decimals

1. Biggest on top

2. Line up decimals; subtract in columns.

3. Borrow by 10’s

4. Strongest wins.

EXAMPLE

Subtract. 283.5 – 3,476.91 - 3476.91 283.5

1.Biggest on top

- 3 4 7 6. 9 1 2 8 3. 5 3. 4 1

2. Line up decimals; subtract in columns

3

- 3 4 17 6. 9 1 2 8 3. 5 3 1 9 3. 4 1

3. Borrow by 10’s. Carry the 1 and leave the 3.

3

- 3 4 17 6. 9 1 2 8 3. 5 - 3 1 9 3. 4 1

3. Biggest one wins.

Section 1.3

16

Multiplication of Decimals

EXAMPLES

29,742 × 538 237,936 892,260 +14,871,000 16,001,196

Next:

2 2 1

29,742 × 30 892,260

Last:

4 3 2 1

29,742 × 500 14,871,000

3. Add the pieces together.

Multiplication of Decimals

1. Multiply each place value 2. Carry by 10’s 3. Add 4. Right size. 1. Add up zeros or decimals

2. Negatives

Start:

7 5 31

29,742 × 8 237,936

Section 1.3

17

Final example with decimals: The only thing left is to count the number of decimal places. We have one in the first number and two in the second. Final answer:

-70139.278

Division of Decimals

-7414.3 × 9.46 444858 2965720 +66728700 -70139278

Next:

1 1 1

74143 × 40 2965720

Last:

3 1 3 2

74143 × 900 66728700

3. Add the pieces together.

4. Right size. Total number of decimal places = 3. Answer is negative.

Start:

2 21

74143 × 6 444858

Division of Decimals

1. Set up. 2. Divide into first. 3. Multiply. 4. Subtract. 5. Drop down. 6. Write answer.

1. Move decimals 2. Add zeros

1. Remainder 2. Decimal

Section 1.3

18

EXAMPLES 5

4298

Step 1. No decimals to set up. Go to Step 2. Step 2.We know that 8 goes into 42 about 5 times.

5

4298

-40

Step 3. Multiply 5×8 Step 4.subtract.

53

4298

-40 29

Step 5. Bring down the 9 to continue on. Repeat steps 2-5 Step 2: 8 goes into 29 about 3 times.

53

4298

-40 29 -24 5

Step 3: Multiply 3×8 Step 4: subtract.

8 doesn’t go into 5 (remainder)

Which means that 429 ÷ 8 = 53 R 5

or in other words 429 ÷ 8 = 5385

Example: 5875 ÷ 22

2

587522

44

Step 2: 22 goes into 58 about 2 times. Step 3: Multiply 2×22 = 44

2

587522

-44 147

Step 4: Subtract. Step 5: Bring down the next column

27

587522

-44 147 154

22 goes into 147 about ???? times. Let’s estimate. 2 goes into 14 about 7 times – try that. Multiply 22×7 = 154 Oops, a little too big

Section 1.3

19

26

587522

-44 147 -132 155

Since 7 was a little too big, try 6. Multiply 6×22 = 132 Subtract. Bring down the next column.

267

587522

-44 147 -132 155 -154 1

22 goes into 155 about ????? times. Estimate. 2 goes into 15 about 7 times. Try 7 Multiply 22×7 = 154. It worked. Subtract. Remainder 1

5875 ÷ 22 = 267 R 1 or 221267

An example resulting in a decimal:

Write 9

4as a decimal:

0000.49 Step 1: Set it up. Write a few zeros, just to be safe.

.4

0000.49

-36 4

Step 2: Divide into first. 9 goes into 40 about 4 times. Step 3. Multiply 4×9 = 36 Step 4. Subtract.

.44

0000.49

-36 40 -36 4

Step 5. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into 40 about 4 times. Step 3: Multiply 4×9 = 36 Step 4: Subtract.

.444

0000.49

-36 40 -36 40 -36 4

Step 5. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into 40 about 4 times. Step 3: Multiply 4×9 = 36 Step 4: Subtract. This could go on forever!

Thus 9

4= .44444. . . which we simply write by .4

The bar signifies numbers or patterns that repeat.

Repeating decimal

Section 1.3

20

Two final examples: 358.4 ÷ -(.005) 296 ÷ 3.1

4.358005.

3584005

Step 1. Set it up and move the decimals

2961.3

00.296031

7

3584005

35

Step 2. Divide into first Step 3. Multiply down

9

00.296031

279

7

3584005

-35 08

Step 4. Subtract Step 5. Bring down

9

00.296031

-279 170

71

3584005

-35 08 - 5 34

Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step 5. Bring down

95.

00.296031

-279 170 -155 150

716

3584005

-35 08 - 5 34 -30 40

Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step 5. Bring down

95.4

00.296031

-279 170 -155 150 -124 26

7168

3584005

-35 08 - 5 34 -30 40 -40 00

Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step 5: Bring down

95.48

000.296031

-279 170 -155 150 -124 260 - 248 120

Section 1.3

21

71680

3584005

-35 08 - 5 34 -30 40 -40 00 - 0 0

Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract

95.483

000.296031

-279 170 -155 150 -124 260 - 248 120 -93 27

-71,680 Step 6: Write answer 95.483 . . .

COMMON MISTAKES

Two negatives make a positive

- True in Multiplication and Division – Since a negative sign simply means opposite direction, when we switch direction twice, we are headed back the way we started.

Example: -(-5) = 5 Example: -(-2)(-1)(-3)(-5) = - - - - -30 = -30

Example: -(-40 ÷ -8) = -(- -5) = -5

- False in Addition and Subtraction – With addition and subtraction negatives and positives work against each other in a sort of tug ‘o war. Whichever one is stronger will win.

Example: Debt is negative and income is positive. If there is more debt than income, then the net result is debt. If we are $77 in debt and get income of $66 then we have a net debt of $11

-77 + 66 = -11 On the other hand if we have $77 dollars of income and $66 of debt, then the net is a positive $11

77 – 66 = 11

One negative in the original problem gives

a negative answer. The decimal obviously keeps going. Round after a couple of decimal places.

Section 1.3

22

Example: Falling is negative and rising is positive. An airplane rises 307 feet and then falls 23 feet, then the result is a rise of 284 feet: 307 – 23 = 284 If, however, the airplane falls 307 feet and then rises 23 feet, then the result is a fall of 284 feet: -307 + 23 = -284 Other examples: Discount is negative and markup or sales tax is positive. Warmer is positive and colder is negative. Whichever is greater will give you the sign of the net result.

1) Percent: Percent can be broken up into two words: “per” and “cent” meaning per hundred,

or in other words, hundredths.

Example: 100

7= .07 = 7%

100

31= .31 = 31%

100

53= .53 = 53%

Notice the shortcut from decimal to percents: move the decimal right two places.

LAWS & PROCESSES

Converting Percents

EXAMPLES

Convert .25 to a percent

.25= 25% Move the decimal two places to the right because we are turning this into a percent

.25=25%

Percents

1. If fraction, solve for decimals. 2. Move decimal 2 places.

3. “OF” means times.

1. Right for decimal to % 2. Left for % to decimal

Section 1.3

23

What is �

�� as a percent?

5 ÷ 32 = .15625 Turn the fraction into a decimal by dividing

.15625=15.625% Move the decimal two places to the right because we are turning this into a percent

5

32= 15.652%

Convert 124% to decimals

124%=1.24 Move the decimal two places to the left because we are turning this into a decimal

124%=1.24

Solving “Of” with Percents The most important thing that you should know about percents is that they never stand alone. If I were to call out that I owned 35%, the immediate response is, “35% of what?” Percents always are a percent of something. For example, sales tax is about 6% or 7% of your purchase. Since this is so common, we need to know how to calculate this. If you buy $25 worth of food and the sales tax is 7%, then the actual tax is 7% of $25.

.07×$25 = $1.75

EXAMPLES

What is 25% of 64?

25%=.25

Turn the percent into a decimal

. 25 × 64 = 16 Multiply the two numbers together 25% of 64 is 16

What is 13% of $25? 13%=.13


. 13 × 25 = 3.25 Multiply the two numbers together 13% of $25 is $3.25

What is 30% of 90 feet?

30%=.30


. 30 × 90 = 27 Multiply the two numbers together

30% of 90 feet is 27 feet

In math terms the word “of”

means multiply.

Section 1.3

24

Section 1.3 Exercises Part A Add. 1. 36,451

+ 2,197 2. 143.29

+ .923 3. 5,834,906.2

+ 54.3227

Subtract. 4. 7- (-2) = 5. -7 – 2 = 6. -13 –(-10) =

7. -18 + 5 = 8. 10 – 57 = 9. -14 – 8 =

10. 234 -57

11. 19.275 -74.63

12. 4,386 -5,119

13. 2.35 -17.986

14. 2,984 - 151

15. Cost:$32.50

Discount:$1.79 Final Price:

16. Temp:67° F Change:18° warmer Final:

17. Altitude: 7,380 ft Fall: 3,200 ft Final:

18. Cost:$32.50

Tax:$2.08 Final Price:

19. Temp: 17° C Change: 28° colder Final:

20. Altitude:300 m Rise:7,250 m Final:

Change into a decimal.

21. 52 22.

41 23.

83

24. 91 25.

87 26.

61

Assignment 1.3a

25

Divide. Example: See examples in section 1.3

27. 2347 28. 1355 29. 58911

30. 3.5604. 31. 428. 32. 2.1511.2

Change into a percent.

33. 129 34.

2019 35.

4515

Using the chart, find out how much money was spent if the total budget was $1600.

36. Insurance 37. House 38. Fun

Find the following:

39. Price: $30.00 Tax rate: 6% Tax:

40. Attendees: 2,300 Percent men: 40% Men:

41. Students: 4 Number of B’s: 3 Percent of B’s:

Preparation. 42. Go to providentliving.org and read the “One for the Money” and “All is Safely Gathered In” pamphlets. Be ready to share thoughts and notes with your group.

Insurance

9%

Car

14%

House

47%

Fun

10%

Food

20%

Expenses

Assignment 1.3a

26

Answers:

1. 38,648 31. 52.5

2. 144.213 32. 72

3. 5,834,960.5227 33. 75%

4. 9 34. 95%

5. -9 35. 33.3%

6. -3 36. $144

7. -13 37. $752

8. -47 38. $160

9. -22 39. $1.80

10. 177 40. 920 men

11. -55.355 41. 75%

12. -733 42. Discuss it together.

13. -15.636

14. 2833

15. $30.71

16. 85° F

17. 4180 ft

18. $34.58

19. -11° C

20. 7550 m

21. .4

22. .25

23. .375

24. .1

25. .875

26. .16

27. 7333 or 33.428571 or 33 R3

28. 27

29. 11653 or 53.54 or 53 R6

30. 1407.5

Assignment 1.3a

27

Section 1.3 Exercises Part B Add. 1. 36,851

+ 3,197 2. 153.29

+ .922 3. 8,434,916.7

+ 54.3527

Subtract. 4. 9 - (-3) = 5. -18 – 32 = 6. -14 –(-19) =

7. -18 + 6 = 8. 15 – 47 = 9. -24 – 8 =

10. 754 -57

11. 29.84 -64.643

12. 4,786 -5,919

13. 2.35 -13.946

14. 23,754 - 4,151

15. Cost:$32.50

Discount:$5.79 Final Price:

16. Temp:67° F Change:28° warmer Final:

17. Altitude: 4,380 ft Fall: 2,230 ft Final:

18. Cost:$33.50



20. Altitude:300 m Rise:2,250 m Final:


21. 54 22.

92 23.

85

24. 81 25.

65 26.

101

Assignment 1.3b

28

Divide.

27. 4347 28. 1356 29. 78912

30. 347.5604. 31. 4536. 32. 12.1251.3


33. 127 34.

2017 35.

3015


36. Food 37. Car 38. Fun

Find the following:




Begin “Budget and Expenses” Portfolio Project 42. Make sure all members of the group have seen the pattern of budget and expense reports found in “All is Safely Gather In” and “One for the Money.” Begin a monthly budget and record of your expenses that will continue through the remainder of the semester. Commit to reporting to your group and receiving reports when all have created a spreadsheet titled, “Budget and Expenses.”

Insurance

9%

Car

14%

House

47%

Fun

10%

Food

20%

Expenses

Assignment 1.3b

29

Answers:

1. 40,048 31. 755

2. 154.212 32. 40.361…

3. 8,434,971.0527 33. 58.3%

4. 12 34. 85%

5. -50 35. 50%

6. 5 36. $260

7. -12 37. $182

8. -32 38. $130

9. -32 39. $4.63

10. 697 40. 1896 men

11. -34.803 41. 91.67%

12. -1,133 42. Submit it to your teacher later in this lesson.

13. -11.596

14. 19,603

15. $26.71

16. 95° F

17. 2150 ft

18. $35.68

19. -21° C

20. 2550m

21. .8

22. .2

23. .625

24. .125

25. .83

26. .1

27. 62

28. 22.5

29. 65.75

30. 1408.675

Assignment 1.3b

30

Section 1.3 Exercises Part C Begin “Budget and Expenses” Portfolio Project.

1. Continue to record all expenses and income for the remainder of the course in a spreadsheet document.

Round the following.


3. Round 385,764,524.83 to the nearest million.


4. 71 × 3250.07 5. 538.9 × 2,892.07 6. 82 × .00000789

Add. 7. 46,821

+ 3,137 8. 756.29

+ .522 9. 8,434.7

+54.3527

Subtract. 10. 115 - (-3) = 11. -19 – 320 = 12. -18 –(-151) =

13. 7.54 -57

14. 298.4 -64.643

15. 3,784 -5,919

16. Cost:$44.50



18. Altitude:300 m Fall:2,250 m Final:


19. 201 20.

94 21.

32

Assignment 1.3c

31

Divide.

22. 4348 23. 1856 24. 68914

25. 347.5602. 26. 5536. 27. 12.17531.


28. 4037 29.

5038 30.

2527


31. Insurance 32. Car 33. Fun

Find the following:




Insurance

8%

Car

15%

Housing

37%

Fun

10%

Food

30%

Expenses

Assignment 1.3c

32

Answers:

1. Titled “Budget and Expenses” and save document on your computer. You will turn it in to your teacher in this lesson.

31. $108.32

2. 54,000 32. $203.10

3. 386,000,000 33. $135.40

4. About 210,000 34. $4.52

5. About 1,500,000 35. 941 men

6. About .00064 36. 73.3%

7. 49,958

8. 756.812

9. 8489.0527

10. 118

11. -339

12. 133

13. -49.46

14. 233.757

15. -2135

16. $47.68

17. 19° C

18. -1950m

19. .05

20. .4

21. .6

22. 54.25

23. 30.83

24. 49.214…

25. 2817.35

26. 921.6

27. 564.903…

28. 92.5%

29. 76%

30. 108%

Assignment 1.3c

33


1) Numerator – the top of a fraction

2) Denominator – the bottom of the fraction

3) Simplify – Fractions are simplified when the numerator and denominator have no factors in

common.

4) One – any number over itself = 1.

5) Common Denominators – Addition and subtraction require like things. In the case of

fractions, “like things” means common denominators.

6) Prime Factorization – Breaking a number into smaller and smaller factors until it cannot be

broken down further.

LAWS & PROCESSES Prime Factorization – One of the ways to get the Least Common Denominator for adding and subtraction fractions that have large denominators is to crack them open and see what they are made of. Scientists get to use a scalpel or microscope. Math guys use prime factorization.

Addition of Fractions

1. Common Denominator

2. Add numerators

3. Carry by denominator

EXAMPLE

Add �

�+

�

�

�

�+

�

�

�

�

Step 1. The least common multiple of 4 and 2 is a 4, so we

replace the �

� with an equivalent fraction, which is

�

� .

3

4

+ 2

4

= 5

Step 2. Now that the denominators are the same, add the numerators.

1. Observation 2. Multiply the denominators 3. Prime factorization

Section 1.4 Fractions

Section 1.4

34

�

�

� �

�

� �

� Step 3. Carry the denominator across.

3

41

2�5

4

Changing from mixed numbers to improper fractions:

5�

�� 5

�

��

��

�

�

��

��

�

Changing them back again:

��

�� 43 � 8 � 5�3 � 5

�

�

Subtraction of Fractions

1. Biggest on top

2. Common Denominator; Subtract numerators

3. Borrow by denominator

4. Strongest wins

EXAMPLE

Do this: �

�� 3

�

�

- 3�

�

5

9

3 �

� is bigger, so put it on top.

- 3 �

�

�

�

5

9

The common denominator is 9,

so change the �

� to a

�

�.

- 2��

�

5

9

- 2�

�

Subtract the numerators. Borrow by denominator as needed.

5

9� 3

1

3� �2

7

9

1. Observation 2. Multiply the denominators 3. Prime factorization

Section 1.4

35

Multiplication of Fractions

EXAMPLES

5

6×

1

3

For multiplication don’t worry about getting common denominators

�

%×

�

�=

�

?

Multiply the numerators straight across

�

%×

�

�=

�

�

Multiply the denominators straight across

5

6×

1

3=

5

18

Multiplication of Fractions

1. No common denominators 2. Multiply Numerators 3. Multiply Denominators

Section 1.4

36

Division of Fractions

EXAMPLES

Divide 3�

$÷

�

�

�×$

$+

�

$÷

�

�

��

$÷

�

�

Turn the fractions into improper fractions

25

7 ×

3

2

Keep the first fraction the same Change the division sign to a multiplication sign Flip the second fraction’s numerator and denominator

��

$×

�

�=

$�

��

Multiply straight across the numerator and denominator

34

7÷

2

3=

75

14

Divide �

�÷ 1

�

�

2

5÷

1 × 4

4+

3

4

2

5÷

7

4

Turn the fractions into improper fractions

2

5×

4

7

KEEP the first fraction the same Change the division sign to a multiplication sign Flip the second fraction’s numerator and denominator

�

�×

�

$=

��

Multiply straight across the numerator and denominator

�

�÷ 1

�

�=

��

Section 1.4

Division of Fractions

1. Improper Fractions

2. Keep it, change it, flip it.

3. Multiply.

37

Now that you have had a little time to multiply fractions together and simplify them, you may have noticed one of the slickest tricks that we can do with fractions, and that is that we can actually do the simplification before we multiply them. Take for example:

10

63×21

55

Now, we can do this the normal way or we can try to notice if there is anything that we will be simplifying out later . . . and do that simplification before we multiply: Normal method:

�

�×

��

=

��

��

and now we try to simplify

��

�� =

�×�× ×�

�×�× ×�×��

which probably took quite a while to get. So,

��

�� =

�×�× ×�

�×�× ×�×��=

�×�× ×�

�×�× ×�×��=

�

��

What I was hoping to show is that the same answer was obtained and the same cancelling was done, but if you are able to see it before you multiply, then you will be able to simplify in a much simpler way. Here is another example:

�

��×

� the 4 and the 8 can simplify before we multiply:

�

��×

��=

��

This may seem like just a convenient way to make the problem go a bit quicker, but it does much more than that. It opens the door to a much larger world. Here is an example. If we travelled 180

miles on 12 gallons of gas, then we calculate the mileage by ��

�� = 15 miles per gallon.

Carrying that example just a bit further, what if gas were $3.2 per gallon? We can actually find how many miles we can drive for one dollar:

��

��×

��

�.�� = 4.7 miles per dollar.

Another example: Carpet is on sale for 15 dollars per square yard. How much is that in dollars per square foot (9 ft2 per yd2)? Now, knowing that we will be able to cancel anything on the top with anything that is the same on the bottom we write the multiplication so the yd2 will cancel out, leaving us with dollars per ft2:

New and improved slick method:

�

�×

��

=

and we try to see if any factors will cancel ahead of time 2×5 3×7

�

�×

��

=

�

��

3×3×7 5×11

Section 1.4

38

�� 1/**.2,

� 31'×

� 31'

# 45'

Then cancel the yd2:

�� 1/**.2,

� 31'×

� 31'

# 45' =

�

� dollars per square foot

= $1.67 per square foot. One more example: A rope costs $15 for 8 feet. How much does is cost per inch? We want to get rid of feet and get inches, so we write the multiplication:

��6 1/**.2,

4++5×

� 4//5

��7 )089+, =

� 1/**.2,

�� )089+, =

$.156 or 15.6 cents per inch. Here are a few numbers that will help you with the conversions: 12 in = 1 foot 16 oz = 1 pound 60 seconds = 1 minute 1000watts = 1 kilowatt

1 yd = 3 ft 60 minutes = 1 hour 1 yd2 = 9 ft2

And also some exchange rates with the American dollar as they were sometime in 2010:

1 Mexican Peso = $0.08 1 Euro = $1.30 1 British Pound = $1.50 1 Brazilian Real = $0.55

1.666

3 00.5

-3 2 0 -18 20 -18

.1562

32 00.5

-3 2 180 -160 200 -192 80

Section 1.4

39

Section 1.4 Exercises Part A Find 4 different names for each fraction:

1. 73 2.

32 3.

117 4.

94

Simplify each fraction.

5. 5236 6.

3627 7.

5616

8. 1210 9.

4515 10.

280120

Create each fraction with a denominator of 36.

11. 61 12.

95 13.

1210

Add or Subtract. Simplify.

14. =+32

52 15. =+

85

41 16. =−

253

307

17. =+127

31 18. =+

65

43 413 19. =−

51

107 39

Example:

11

3 ...

000,11

000,3,

110

30,

55

15,

44

12,

33

9,

22

6,

11

3

Example:

1413

146

147

73

21

=+

=+

Example:

41

83 135 −

-4113

835

-8213

835

-8

1012

835

-877

Swap to subtract. Answer is negative

Common denominator Borrow from the 13.

Assignment 1.4a

40

20. =−76

149 63 21. =+

32

72 94 22. =−

43

85 912

Fill out the table.

Mixed Improper

23. 987−

24. 513

25. 843

26. 451

Find the multiplicative inverse or reciprocal of each number.

27. 74 28.

92 29. -

107 30.

87

31. -65 32. 13 33.

4213 34.

37

Divide.

35. =÷31

52 36. =÷

83

41 37. =÷

83

65

38. =÷127

83 39. =÷

61

43 72 40. =÷

32

75 35

41. =÷109

547 42. =÷

32

87 9 43. =÷

83

612

Preparation. 44. If you drive 280 miles on 12 gallons of gas, how many miles per gallon do you get? 45. If you drive 280 miles on 12 gallons of gas, and gas is $3.20 per gallon, how many miles per dollar do you get?

Example:

85

58

Example:

=÷54

832

=×45

832 Multiply by reciprocal

=×45

819 Change to improper fraction

=×45

819

3295 or

32312 Multiply straight across.

Assignment 1.4a

41

Answers:

1. ...,,,,4921

2812

219

146 others 31. 5

6− or 511−

2. ...,,,,1812

1510

96

64 others 32. 13

1

3. ...,,,,5535

4428

3321

2214 others 33. 13

42 or 1333

4. ...,,,,6328

3616

2712

188 others 34. 7

3

5. 139 35. 5

6 or 511

6. 43 36. 3

2

7. 72 37. 9

20 or 922

8. 65 38. 14

9

9. 31 39. 86

33

10. 73 40. 77

120 or 77431

11. 366 41. 3

26 or 328

12. 3620 42. 232

21

13. 3630 43. 9

52 or 975

14. 1511 or

1516 44. Discuss it together.

15. 87 45. Discuss it together.

16. 15017

17. 1211

18. 12718

19. 216

20. 1433−

21. 212013

22. 872

23. 971−

24. 516

25. 835

26. 12�

�

27. 47 or

431

28. 29 or

214

29. 710− or

731−

30. 78

Assignment 1.4a

42

Section 1.4 Exercises Part B Create each fraction with a denominator of 24.

1. 32 2.

127 3.

4840


4. =+72

52 5. =+

97

43 6. =−

87

125

7. =−97

32 163 8. =+

65

75 67 9. =−

53

85 92

Fill out the table.

Mixed Improper

10. 952−

11. 746

12. 8

35

13. 1157

Find the multiplicative inverse or reciprocal of each number.

14. 53 15.

943 16. -

125 17. 7

Divide.

18. =÷35

72 19. =÷

76

43 20. =÷

94

61

21. =÷103

322 22. =÷

21

85 4 23. - =÷

75

732


24. 52 25.

41 26.

83

27. 91 28.

87 29.

61

Assignment 1.4b

43

Change into a fraction and simplify.

30. .5 31. .7 32. .45

33. .52 34. .75 35. .6

Convert the following units. Example: Dog food cost $7.00 for 20 pounds. How many ounces per dollar?

Solution: ��

�� ×

��

��=

��

�� =45.71 ounces per dollar

36. Cereal cost $4.50 for 2 pounds. How much did it cost per ounce?

37. Fishing line costs $.02 per foot. How much would 200 yards cost?

38. I was able to drive 250 miles on 15 gallons of gas. If gas costs $3.10 per gallon, how many

miles can I drive per dollar?

39. If my sprinkler sends out 5 gallons per minute, and if water costs $0.65 per 1000 gallons,

how much does watering my lawn cost per hour?

40. How many Pesos are equal to 5 Euros? (1 Mexican Peso = $0.08, 1 Euro = $1.30)

41. How many Reals are equal to 7 Pounds? (1 Brazilian Real = $0.55, 1 British Pound = $1.50)

42. Create a visual chart for all arithmetic of decimals. Use plenty of examples.

43. Create a visual chart for all arithmetic of fractions including Unit Conversions.

Example .12

.12 = 12 (100th) = 10012 =

253 simplify

Assignment 1.4b

44

Answers:

1. 2416 31. 10

7

2. 2414 32. 20

9

3. 2420 33. 25

13

4. 3524 34. 4

3

5. 36191 or

��

�% 35. 5

3

6. 2411− 36. $0.14 per ounce

7. 9113− 37. $12.00

8. 422314 38. 5.38 miles per dollar

9. 40396− 39. $0.20 per hour

10. 923− 40. 81.25 Pesos

11. 746 41. 19.09 Reals

12. 834 42. Part of Portfolio

13. 1125 43. Part of Portfolio

14. 35

15. 319

16. 512−

17. 71

18. 356

19. 87

20. 83

21. 988

22. 365

23. -523

24. .4

25. .25

26. .375

27. .1

28. .875

29. .16

30. 21

Assignment 1.4b

45

Section 1.4 Exercises Part C Exam 1 Review Exercises

Estimate the product (round to the greatest value, then multiply). 1. 2,589,000×59.34 2. .005608×.07816 3. 3.847×2,564

Add. 4. 36,841

+ 249.7 5. 723.3

+ 39.7 6.

1413

149 516 + =

Subtract. 7. Temp: -35.5° F

Change: 13.4° warmer Final:

8. -8 – (-11) = 9. =−76

74 113

Multiply. 10. Cost: $35.20

Quantity: 17 Total:

11. 369×(-23) = 12. =×1211

54

Add or Subtract. Simplify. 13. =+

95

23 14. =+

145

1211 15. =−

65

185

16. =−97

61 615 17. =+

81

109 135 18. =−

142

94 912

Fill out the table.

Mixed Improper

19. 753−

20. 6

59

Divide.

21.

185

98

22. =÷

32

98 4 23. ( ) =−÷

5

4

4

37


24. 125 25.

97 26.

72

Change into a fraction and simplify.

27. .3 28. .055 29. .375

Assignment 1.4c

46

Divide.

30. 4857 31. 7813 32. 67343

33. 31.475. 34. 4.5620004.

35. A dishwasher uses about 1400 watts of power. If the power company charges 9 cents per kilowatt-hour, how much does it cost to run a dishwasher for 16 hours in the month? 36. I bought 8 yards of rope for $9.84. How much did it cost per foot? Change into a percent.

37. 2524 38.

4036 39.

5017


40. Car 41 House 42 Food

Find the following:

43. Price: $45.20 Tax rate: 7% Tax: Final Price:

44. Attendees: 239 Percent men: 29% Men:

45. Price: $15.30 Discount: 30% Amount of discount: Final Price:

46. Round to the nearest ten: 583.872

Insurance

9%

Car

14%

House

47%

Fun

10%

Food

20%

Expenses

Assignment 1.4c

47

Answers:

1. About 180,000,000 31. 31260 or 260.3

2. About .00048 32. 432815 or 15.65116…

3. About 12,000 33. 94.62

4. 37,090.7 34. 1,406,000

5. 763 35. $2.02

6. 7422 36. $0.41 per foot

7. -22.1° F 37. 96%

8. 3 38. 90%

9. 7511 39. 34%

10. $598.40 40. $448

11. -8,487 41. $1504

12. 1511 42. $640

13. 1812 43. $3.16, $48.36

14. 84231 44. 69 men

15. 95− 45. $4.59, $10.71

16. 1878 46. 580

17. 40119

18. 63193

19. 726−

20. 659

21. 513 or

516

22. 214

23. -16119 or -

16155

24. .416

25. .7

26. .285714

27. 103

28. 20011

29. 83

30. 7269 or 69.285714

Assignment 1.4c

48

Chapter 2:

CALCULATORS and FORMULAS

Overview 2.1 Exponents and Calculator Usage 2.2 Variables and Formulas 2.3 More Variables and Formulas - Excel

49

While we are on multiplication, did you know that there is some short hand? Remember

when we started multiplication we did: 6+6+6+6+6+6+6+6+6 = 54 but we did it a bit shorter 9 9×6 = 54 There is a way to write multiplication in shorthand if you do the same thing over and over again: 2×2×2×2×2×2×2 = 128 7 For the shorthand we write 27 = 128. That little 7 means the number of times that we multiply 2 by itself and is called and exponent; sometimes we call it a power. Here are a couple more examples: 53= 125 72 = 49 24 = 16 Pretty slick. You won’t have to memorize them . . . yet, but you should be familiar enough with them to be able to recognize them. Some of the easiest to calculate are the powers of 10. Try these: 104= 10,000 108 = 100,000,000 103 = 1,000

EXAMPLE

Evaluate 74

7� = 7 × 7 × 7 × 7 49 × 7 × 7 343 × 7 2401 Answer: 2401

Set up the bases, and then multiply each couple in turn.

Section 2.1 Exponents

Section 2.1

50

Order of Operations

The last small note to finalize all your abilities in arithmetic is to make sure you know what you need to do when you have multiple operations going on at the same time. For example, 2 + 3 × 4 – 5 If you were to read that from left to right you would first add the 2 and the 3 to get 5 and then multiply by 4 to get 20 and then subtract 5 to get 15. Unfortunately, that doesn’t jive with what we have learned about what multiplication is. Remember that multiplication is a shorthand way of writing repeated addition. Technically we have: 2 + 3 × 4 – 5 = 2 + 4 + 4 + 4 – 5 = 9. Ahh, now there is the right answer. It looks like we need to take care of the multiplication as a group, before we can involve it in other computations. Multiplication is done before addition and subtraction. Here is another one: 4 × 32 – 7 × 2 + 4 Now remember that exponents are shorthand for a bunch of multiplication that is hidden, so we need to take care of that even before we do multiplication: 4 × 32 – 7 × 2 + 4 = Take care of exponents 4 × 9 – 7 × 2 + 4 = Take care of multiplication 36 – 14 + 4 = Add/Sub left to right. 22 + 4 = 26. Now division can always be written as multiplication of the reciprocal, so make sure you do division before addition and subtraction as well. Look at that. We have established an order which the operations always follow, and we need to know it if we are to get the answers that the problem is looking for: 1st – Exponents 2nd – Multiplication and Division (glues numbers together) 3rd – Addition and Subtraction (left to right) Parentheses can change everything. We put parentheses when we intend on grouping (or gluing) numbers together manually. Though they all have the same numbers and operations, see the difference between these: 52542

23632

2632 2

−=−

=÷×−

=÷×−

( )

1601622

23242

2182

26322

2

−=−

=÷−

=÷−

=÷×−

( )

18236

2361

2632 2

−=÷−

=÷×−

=÷×−

( )

( )

( )

1282256

216

2182

2632

2

2

2

=÷

=÷−

=÷−

=÷×−

Section 2.1

51

Section 2.1 Exercises Part A Calculator Usage Assignment

On this assignment, you should use your calculator. Become familiar with it. It is now your friend! Estimate the product (round to the greatest value; then multiply).

1. 75,800×49.34 2. .004208×.06916 3. 4.447×7,164

Add. 4. 37,291

+ 348.23 5. 5.871

+ 39.7 6.

23

5

23

9 517 + =

Subtract. 7. Temp: 85.3° F

Change: 130.4° colder Final:

8. -5 –3 = 9. =−118

114 1523


Quantity: 27 Total:

11. 441×29 = 12. - =×1116

52

Find. 13. 37= 14. 272= 15. 117=


16. =+94

43 17. =+

107

85 18. =−

97

158

19. =−94

81 714 20. =+

81

109 195 21. =−

163

85 54

Fill out the table. Mixed Improper

22. 1147

23. 25−

Divide. 24.

187

1211

25. =÷

21

65 4 26. =÷

83

857


27. 117 28.

53 29.

92

Assignment 2.1a

52

Change into a fraction and simplify. 30. .07 31. .44 32. .625

Divide.

33. 3437 34. 796 35. 627357

36. 731.45. 37. 4.967004.

Evaluate 38. 5 − 3� + 8 ÷ 2 39. (5 − 3)� + 8 × 2 40. 5 − (4� + 8) ÷ 2

41. Change 60 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent.

42. 3028 43.

5741 44.

10037


45. Fun 46. Insurance 47. Food

Find the following:


49. Attendees: 48 Percent kids: 25% Kids:

50. Students: 30 Number of A’s: 24 Percent of A’s:

Insurance

9%

Car

16%

House

45%

Fun

13%

Food

17%

Expenses

Assignment 2.1a

53

Answers:

1. About 4,000,000 31. 2511

2. About .00028 32. 85

3. About 28,000 33. 49

4. 37,639.23 34. 6113 or 13.16

5. 45.571 35. 110.0526…

6. 231422 36. 9.462

7. -45.1° F 37. 241,850

8. -8 38. 0

9. 1177 39. 20

10. $1,036.80 40. -7

11. 12,789 41. 88 feet per second

12. -5532 42. 93.3%

13. 2187 43. 71.9%

14. 729 44. 37%

15. 19,487,171 45. $316.81

16. 3671 46. $219.33

17. 40131 47. $414.29

18. 4511− 48. $26.64, $407.14

19. 72496 49. 12 kids

20. 40125 50. 80%

21. 169−

22. 1181

23. 212−

24. 1452 or

1433

25. 275

26. 3120 or

361

27. .63

28. .6

29. .2

30. 1007

Assignment 2.1a

54

Section 2.1 Exercises Part B Add. 1. 57,831

+ 348.23 2. 4.83

+ 39.7 3.

115

119 814 + =

Subtract. 4. Temp: -85.3° F

Change: 130.4° colder Final:

5. -5 –53 = 6. =−218

214 1523


Quantity: 527 Total:

8. - =× 1415

52

Find. 9. 35 = 10. 372 = 11. (5.8)3 =

12. (2.38)2 = 13. (1.07)27 = 14. (1.12)12 =

15. If I place 2 cents on the first square of a chess board, 4 cents on the second square, and keep doubling the amount on each square, how much money will be on the 30th square? Fill out the table.

Mixed Improper

16. 325

17. 2

57−

18. A product costs $7 for 20 pounds. How much is that in cents per ounce? 19. Change 17 Euros into pesos. (1 Mexican Peso = $0.08, 1 Euro = $1.30) 20. Change 60 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent.

21. 3524 22.

6472 23.

200014

Using the percentages, find out how much money was spent if the total budget was $2437.

24. Fun – 12.3% 25. Insurance – 7.9% 26. Food – 38%

Find the following:



29. Students: 250 Number of A’s: 147 Percent of A’s:

Assignment 2.1b

55

Final Price:

30. For a savings account that begins with $100 and has a 5% interest rate, fill out the following table:

Time Beginning Balance Interest earned Ending Balance

1st year 100 .05 × 100 = 5 105

2nd year 105 .05 × 105 = 5.25 110.25

3 110.25 .05 × 110.25 =5.51 115.76

4 115.76

5

6

7

8

9

10

11

12

31. For a savings account that begins with $100 and has a 6% interest rate, fill out the following table:

Time Beginning Balance Ending Balance

1st year 100 100 × 1.06 = 106

2nd year 106 106 × 1.06 = 112.36

3 112.36 112.36 × 1.06 = 119.10

4 119.10

5

6

7

8

9

10

11

12

32. Discuss in your group why multiplying by .05 and then adding to the balance is the same as multiplying the balance by 1.05. 33. If a savings account started at $100 and earned 7% per year, how much would be in the account at the end of 12 years? 34. If a savings account started at $100 and earned 7% per year, how much would be in the account at the end of 22 years? 35. How can exponents be used to find the balance after many years?

Assignment 2.1b

56

1. 58,179.23 31. 12 year end balance - $201.22 ($201.23 also acceptable)

2. 44.53 32. 1 adds in the beginning balance and .05 adds in the 5%

3. 23�

�� 33. $225.22

4. -215.7° F 34. $443.04

5. -58 35. #34 can be done by 100 × (1.07)22

6. 7��

��

7. $20,236.80

8. - �

�

9. 243

10. 1,369

11. 195.112

12. 5.6644

13. 6.214

14. 3.896

15. $10,737,418.24

16. ��

�

17. - 28�

�

18. 2.19 cents per ounce

19. 276.25 Pesos

20. 88 feet per second

21. 68.6%

22. 112.5%

23. 0.7%

24. $299.75

25. $192.52

26. $926.06

27. $33.64; $514.14

28. 97 kids

29. 58.8%

30. 12 year end balance - $179.59 ($179.60 also acceptable)

Assignment 2.1b

57

Section 2.1 Exercises Part C 1. Find three different places to save your money. Report the interest rates to your group, and receive their reports. Find.

2. 45= 3. 872= 4. (2.7)5=

5. (5.38)2 6. (1.06)25

7. (1.11)13

Fill out the table. Mixed Improper

8. 525

9. 3

37−

10. If I place 1 cent on the first square of a chess board, 2 cents on the second square, and keep doubling the amount on each square, how much money will be on the 20th square? 11. A product sells for $2.50 per square foot. How much is that per square yard? 12. Change 400 Pesos into Pounds. (1 Mexican Peso = $0.08, 1 British Pound = $1.50) 13. Change 50 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent.

14. ��

�� 15. ��

�� 16. �

��,��

Using the percentages, find out how much money was spent if the total budget was $287.

17. Fun – 17.3% 18. Insurance – 6% 19. Food – 84%

Find the following:



Assignment 2.1c

58

22. For a savings account that begins with $350 and has a 5% interest rate, fill out the following table and place the entries in the “Life Plan” spreadsheet on Sheet 2:


1st year 350 350 × 1.05 = 367.50

2nd year 367.50

3

4

5

6

7

8

9

10

11

12

23. If a savings account started at $300 and earned 7% per year, how much would be in the account at the end of 22 years? 24. For a savings account that begins with $100 and has a 6% interest rate and to which you are able to add $25 per year, fill out the following table and place it on Sheet 2 of your Life Plan spreadsheet:


1st year 100 100 × 1.06 + 25 = 131

2nd year 131 131 × 1.06 + 25 = 163.86

3 163.86 163.86 × 1.06 + 25 =

4

5

6

7

8

9

10

11

12

25. If a savings account started at $200 and earned 7% per year, how much would be in the account at the end of 12 years if you are able to add $40 per year?

Assignment 2.1c

59

1. Complete when all reports are done.

2. 1024

3. 7569

4. 143.489

5. 28.944

6. 4.29

7. 3.88

8. ��

�

9. −12�

�

10. $5,242.88

11. $22.50 per square yard

12. 21.33 pounds

13. 73.3 feet per second

14. 66.7%

15. 54.4%

16. .02%

17. $49.65

18. $17.22

19. $241.08

20. $5.63; $86.03

21. (135.8) 136 kids

22. $628.55

23. $1329.12

24. $622.97

25. $1,165.98

Assignment 2.1c

60

Variables and Formulas


1) Variables: These symbols, being letters, actually represent numbers, but the numbers can

change from time to time, or vary. Thus they are called variables.

Example: Tell me how far you would be walking around this rectangle. 24 ft 15 ft 15ft 24 ft It appears that to get all the way around it, we simply add up the numbers on each side until we get all the way around. 24+15+24+15 = 78. So if you walked around a 24ft X 15ft rectangle, you would have completed a walk of 78 ft. I bet we could come up with the pattern for how we would do this all of the time. Well, first of all, we just pick general terms for the sides of the rectangle: length width width length Then we get something like this: Distance around the rectangle = length + width + length + width Let's try and use some abbreviations. First, “perimeter” means “around measure”. Substitute it in: Perimeter = length + width + length + width Let's go a bit more with just using the first letters of the words: P = l + w + l + w

Notice now how each letter stands for a number that we could use. The number can change from time to time. This pattern that we have created to describe all cases is called a formula.

Section 2.2

Section 2.2 Variables and Formulas

61

2) Formula: These are patterns in the form of equations and variables, often with numbers,

which solve for something we want to know, like the perimeter equation before, or like:

Area of a rectangle: A = B × H

Volume of a Sphere: � =�

��

Pythagorean Theorem: �� + �� = �� Through the same process we can come up with many formulas to use. Though it has all been made up before, there is much to gain from knowing where a formula comes from and how to make them up on your own. I will show you on a couple of them. Distance, rate If you were traveling at 40mph for 2 hours, how far would you have traveled? Well, most of you would be able to say 80 mi. How did you come up with that? Multiplication: (40)(2) = 80 (rate of speed) ⋅ (time) = distance or in other words:

rt = d

where r is the rate t is the time d is the distance Percentage If you bought something for $5.50 and there was an 8% sales tax, you would need to find 8% of $5.50 to find out how much tax you were being charged. .44 = .08(5.50) Amount of Tax = (interest rate) ⋅ (Purchase amount) or in other words:

T = rP Where T is tax r is rate of tax P is the purchase amount. Interest This formula is a summary of what we did in the last section with interest. If you invested a principal amount of $500 at 9% interest for three years, the amount in your account at the end of three years would be given by the formula: A = 500(1.09)3 = $647.51

Section 2.2

62

A = P(1 + r)Y where A is the Amount in your account at the end P is the principal amount (starting amount) r is the interest rate Y is the number of years that it is invested. Temperature Conversion Most of us know that there is a difference between Celsius and Fahrenheit degrees, but not everyone knows how to get from one to the other. The relationship is given by:

C = 95 (F – 32)

where F is the degrees in Fahrenheit C is the degree in Celsius Money If you have a pile of quarters and dimes, each quarter is worth 25¢ (or $.25) and each dime is worth 10¢ ($.10), then the value of the pile of coins would be:

V = .25q + .10d where V is the Total Value of money q is the number of quarters d is the number of dimes 3) Common Geometric Formulas: Now that you understand the idea, these are some basic

geometric formulas that you need to know:

l

w Rectangle

P = 2l + 2w

A = lw

P is the perimeter l is the length w is the width A is the Area

Section 2.2

63

b a h

Parallelogram

P = 2a + 2b A = bh

P is the perimeter a is a side length b is the other side length A is the Area h is the height

b a h d B

Trapezoid

P = b+a+B+d

A = 21 h(B+b)

P is perimeter b is the shorter base B is the longer base a is a leg d is a leg A is the Area h is the height

h b

Triangle

P = s1+s2+s3

A = 21 bh

P is the perimeter s is a side A is the Area b is the base h is the height

b c a

Triangle

a + b + c = 180°

a is one angle b is another angle c is another angle

Section 2.2

64

H w l

Rectangular Solid

SA = 2lw+2wh+2lh

V = lwh

SA is the Surface Area l is the length w is the width h is the height V is volume

r

Circle

C = 2πr A = πr2

C is the Circumference or Perimeter π is a number, about 3.14159 . . . it has a button on your calculator r is the radius of the circle A is the area inside the circle.

r h

Cylinder

LSA = 2πrh

SA =2πrh+2πr2 V = πr2h

LSA is Lateral Surface Area or area just on the sides π is a number, about 3.14159 . . . it has a button on your calculator r is the radius of the circle h is the height SA is total surface area V is Volume

Section 2.2

65

h l r Cone

LSA = πrl

SA = πr2+ πrl

V = 31πr2h

LSA is Lateral Surface Area or the area just on the sides π is a number, about 3.14159 . . . it has a button on your calculator r is the radius of the circle h is the height l is the slant height SA is total surface area

r Sphere

SA = 4πr2

V =34πr3

SA is the surface area π is a number, about 3.14159 . . . it has a button on your calculator r is the radius V is the Volume

Section 2.2

66

Section 2.2 Exercises Part A Add or Subtract. Simplify.

1. =−83

87 136 2. =+

43

125 1877 3. =−

152

65 9721

Divide.

4. 574.97.3 5. 7.2546000 6. 65.37008.

7. If a wood floor costs $4.50 per square foot, how much is that per square yard?

8. How much does it cost to run a 700 watt microwave for 17 hours if the power company charges 12 cents per kilowatt-hour?

Find the following:

9. Price: $39.48 Tax rate: 5% Tax: Total Price:

10. Price: $2,736.00 Percent off: 35% Amount saved: Final Price:

11. Birds: 140 Black : 47 Percent of black birds:

Evaluate the following:

12. )9(834 −⋅ 13. pd ⋅⋅⋅ 75 14. )45(2)93(5 3+⋅−−

15. 45(7.8) 16. 273 ⋅⋅⋅ m 17 2(32)+5(4)+8 ⋅ m

Find the perimeter of the following shapes:

18. 17 11 19 t+3 8

19. 14 19 k-12

20. 15 9 5 13 r

Evaluate the following when m = 3, n = 7, t = 15, and a = 4.

21. 3t - 7 22. 2(n+9) 23. a

283 ⋅ + m2

Assignment 2.2a

67

24. 12 – a3 25. m – 12 26. 2n – 3a + 5t

Use the formula for distance, rate and time to calculate the distance.

27. r = 7 t = 15 d =

28. r = 55 t = 7.2 d =

29. r = 45

t = 312

d =

Use the formula for angles in a triangle to calculate the measure of the remaining angle.

30. a = 73° b = 24° c =

31. a = 38° b = c = 59°

32. a = b= 24° c= 48°

Use the formulas for Money totals (you may have to make up your own) when q stands for quarters (1 quarter = $0.25), d for dimes (1 dime = $0.10), n for nickels (1 nickel = $0.05) and p for pennies (1 penny = $0.01).

33. q = 9 d = 12 V =

34. p = 19 d = 17 V =

35. n = 37 q = 23 V =

Use the formulas for Temperature Conversion.

36. F = 75° C =

37. F = 15° C =

38. F = -23° C =

Preparation: 39. If the formula for area of a circle is A=πr2 What is the area of a circle with radius 7? 40. Where did π come from? (Try finding out using dictionaries or the internet)

Example: r = 3 t = 14 d =

Formula is found in section 2.3: rt = d 3(14) = d 42 = d

Assignment 2.2a

68

Answers:

1. 216− 31. 83°

2. 61195 32. 108°

3. 10375− 33. $3.45

4. 2.5876 34. $1.89

5. .04245 35. $7.60

6. 4,706.25 36. 23.9° C

7. $40.50 per square yard 37. -9.4° C

8. $1.43 38. -30.6° C

9. $1.97 and $41.45 39. Discuss together.

10. $957.60 and $1,778.40 40. Discuss together.

11. 33.6%

12. -60

13. 35dp

14. -102

15. 351

16. 42m

17. 38 + 8m or 8m + 38

18. 58 + t or t + 58

19. k + 21 or 21 + k

20. 42 + r or r + 42

21. 38

22. 32

23. 30

24. -52

25. -9

26. 77

27. 105

28. 396

29. 105

30. 83°

Assignment 2.2a

69


Evaluate the following when p = 8, r = -7, t = 32 , and a = 3.

1. 12 + a3 2.

3r12 - 10a 3. 5r – 7p + 6t

Use the formula for Interest to calculate the amount in the account at the end of the time period.

4. P = 520 r = 6.2% Y = 4 A =

5. P = 35,000 r = 6% Y = 9.3 A =

6. P = 200 r = 8.9% Y = 7 A =

Use the formulas for Money totals (you may have to make up your own) when q stands for quarters, d for dimes, n for nickels and p for pennies.

7. q = 25 d = 17 n = 15 V =

8. p = p d = q-13 V =

9. p = p q = q n = q+7 V =

Use the formula for Temperature Conversion to calculate the temperature in degrees Celsius.

10. F = 300° C =

11. F = -45° C =

12. F = 102° C =

Use the formulas for a cone to calculate the missing value.

13. r = 6 h = 11 V =

14. r = 9 l = 5 SA =

15. r = 3 l = 8 LSA =

Use the formulas for a triangle to calculate the missing value.

16. b = 24 h = 5 A =

17. b = 15 h = 4 A =

18. Two angles are 37° and 81°; what is the third?

Use the formulas for a trapezoid to calculate the missing value.

19. b = 7 B = 10 h = 7

20. b = 9 B = 15 h = 3

21. b = 7 B = 15 a = 12

Assignment 2.2b

70

A = A= d = 8 P =

Use the formulas for a rectangular solid to calculate the missing information.

22. l = 6 w = 9 h = 7 SA =

23. l = 4 w = 15 h = 7 SA =

24. l = 6 w = 14 h = 2 V =

Simplify.

25. 8y + 5y 26. 4a – 9 + 4a 27. 16r – 5t + 3t + 12r

28. 7(x – 5) +15x 29. 7t + 4(t + 12) 30. 8 – 6(7 – 4t) +4t

31. 8 – 12x2 + 5 + 3x2 32. 7x2 – 5x – 9x2 + 13x 33. 13xy + 7x(6y – 4)

As a group, discuss the following: 34. If the radius and height in #13 are in meters, what is the unit of the Volume? 35. If the bases and height in #19 are in inches, what is the unit of the Area? 36. If all the sides in #21 are measured in millimeters, what is the unit of the Perimeter? 37. If the radius and height in #15 are in miles, what is the unit of the Lateral Surface Area? 38. If all the sides in #24 are measured in yards, what is the unit of the Volume?

Assignment 2.2b

71

Answers:

1. 39 31. -9x2 + 13

2. 76− 32. -2x2 + 8x

3. -87 33. 55xy – 28x

4. $661.46 34. m3 – cubic meters

5. $60,174.51 35. in2 – square inches

6. $363.27 36. mm – millimeters

7. $8.70 37. mi2 – square miles

8. V = .01p + .1(q-13) 38. yd3 – cubic yards

9. V = .01p + .3q + .35

10. 148.9°

11. -42.8°

12. 38.9°

13. 132π or 414.69

14. 126π or 395.84

15. 24π or 75.4

16. 60

17. 30

18. 62°

19. 59.5

20. 36

21. 42

22. 318

23. 386

24. 168

25. 13y

26. 8a – 9

27. 28r – 2t

28. 22x – 35

29. 11t + 48

30. 28t – 34

Assignment 2.2b

72

Section 2.2 Exercises Part C Please label everything with the correct units.

Evaluate the following when f = 5, r = -7, t = 32 , and a = -2.

1. 6t – f3 2. +2f

12 - 10a t 3. 2fr – 31a + 15a

Use the formula for Interest.

4. P = $15,000 r = 6.2% Y = 7 A =

5. P = £ 2,300 r = 6% Y = 8.7 A =

6. P = € 1,300 r = 8.9% Y = 7 A =


7. q = t+5 d = m n = 13 V =

8. p = 15 d = 9 V =

9. p = h+9 q = 7 n = x - 20 V =

Use the formula for Temperature Conversion to calculate the temperature in degrees Celsius.

10. F = -20° C =

11. F = 59° C =

12. F = 32° C =

Use the formulas for a cylinder to calculate the missing value.

13. r = 6 in h = 12 in V =

14. r = 9 m h = 5 m SA =

15. r = 3 yd h = 8 yd LSA =

Use the formulas for a triangle to calculate the missing value.

16. b = 6 ft h = 5 ft A =

17. b = 15 cm h = 4 cm A =

18. Two angles are 45° and 79°; what is the third?

Use the formulas for a trapezoid to calculate the missing value.

19. b = 9 km B = 11 km h = 7 km A =

20. b = 8 mm B = 15 mm h = 5 mm A=

21. b = 12 ft B = 25 ft a = 13 ft d = 17 ft P =

Assignment 2.2c

73

Simplify.

22. 9y – 11y 23. 10a – 2b + 4a – 9b 24. 8(r – 7t) + 8(t +6r)

25. 2(x – 5) +7 26. 8m+ 4(m + 15t) 27. 9 – 5(6 – 9p) +4p

28. 8x2 – 34x3 + 9x2 + 10x3 29. 12x4 – 5x – 4x4 + 13x 30. 3xy – 7x(5y – 4m)

31. If tile costs $1.50 per square foot, how much is that per square yard?

32. How much does it cost to run an 800 watt microwave for 17 hours if the power company charges 11 cents per kilowatt-hour?

33. Change 3 Euros into Pesos. (1 Euro = $1.30, 1 Mexican Peso = $0.08)

34. Change 66 feet per second into miles per hour. (5280 feet = 1 mile)

Assignment 2.2c

74

Answers:

1. -121 31. $13.50 per yd2

2. 1538− or

1582− 32. $1.50

3. -38 33. 48.75 Pesos

4. $22,854.03 34. 45 miles per hour

5. £ 3,818.47

6. € 2,361.23

7. .25t + .1m + 1.9

8. $1.05

9. .01h + .05x + .84

10. -28.9° C

11. 15° C

12. 0° C

13. 1,357.17 in3

14. 791.68 m2

15. 150.8 yd2

16. 15 ft2

17. 30 cm2

18. 56°

19. 70 km2

20. 2115 or 57.5 mm2

21. 67 ft

22. -2y

23. 14a – 11b

24. 56r – 48t

25. 2x – 3

26. 12m + 60t

27. 49p – 21

28. -24x3+17x2

29. 8x4 + 8x

30. -32xy + 28xm

Assignment 2.2c

75

Section 2.3 – More Formulas A calculator is a beautiful thing. You have been able to use one for a short time now and have probably enjoyed it considerably when compared to doing all of the math by hand. You are now ready to take another step with a much more powerful calculator – a computer. During this lesson, you are going to learn the basics of spreadsheets and how to make a computer do the calculations for you.

During this discussion, we will use Microsoft Excel as the spreadsheet, but similar functions can be done in spreadsheets that are available at no cost such as OpenOffice – Calc.

Microsoft Excel Basics Microsoft Excel is spreadsheet software that allows you to perform calculations that help solve math problems in this course. You supply key figures and Excel automatically makes the calculations for you. Open Excel on your computer by clicking Start then Programs then Microsoft Excel. The main spreadsheet in Excel will appear. The spreadsheet is divided into cells each of which has a column and row address. Excel identifies columns by alphabetical letters and rows by numbers. The first cell in the upper left corner is A1. The cell to the right of it is B1 and so forth. The cell below A1 is A2 and so forth. You enter numbers, formulas, or words into the cells. Use the following guidelines as you enter data into Excel.

• It is easiest to enter numerical data in cells by using the number keypad on your keyboard. Be sure the Num Lock key is pressed and the Num Lock light is on.

• The number keypad also has four arithmetic functions you will need which are + (add), - (subtract), * (multiply), and / (divide). It also has the numbers and an enter key so you can enter data rapidly using the keypad.

• Enter the = (equal) sign in the cell before you perform any calculation in Excel. This tells Excel you want it to perform a calculation.

Use the following guidelines to format data in Excel.

• Never enter dollar signs ($) or commas (,) when entering data in Excel. Enter these by formatting the cell.

• Right click the cell or range of cells and select Format Cells. This opens a window that allows you to set the format in number, general, currency, percent, etc. You can set the number of decimal points you want to use and you can set alignment, font, etc. in this window. The cell format already has been set in most of the exhibits you will be using in this course.

TIP: You can also format data in cells by clicking the cell or range of cells then clicking the

appropriate symbol on the formatting tool bar.

Section 2.3

76

Lifelong Income Example – Beginning Salary You can estimate your lifelong income using Excel To determine Lifelong Income do the following: 1. Enter the beginning hourly rate you will earn in your first job after you graduate in cell E3,

for example $15.00. 2. Enter the number of hours you will work in a year in cell E5 as follows: =40*52 where 40 is

the number of hours per week and 52 is the number of weeks in a year. 3. Press enter. Excel automatically multiplies 40 hours per week times 52 weeks per year and

provides the result or 2080 working hours per year. 4. To calculate your first year salary in cell E7, enter (a) the equal sign, (b) click cell E3 (rate

per hour) then enter * (multiplication sign) and (c) click cell E5 (hours per year). 5. Press enter. Excel calculates your first year’s income at $31,200. These entries are illustrated

below:

Yearly Income Calculation – Format In Cell Enter Results

Rate Per Hour: E3 15 $15.00

Hours Per Year E5 =40*52 2080

Income - First Year of Employment (Beginning): E7 =E3*E5 $31,200.00

When you click on a cell that has a calculation set up, the formula for that cell appears in the formula line (to the right of the = sign) at the top of the page. For example, the formula line for the calculation performed in step 5 above would be: =E3*E5 Once your calculations are in place, Excel can save you time and effort if changes are required. If you were to change the beginning rate per hour to $10.00 and you have used the cell addresses in each of your formulas, Excel will recalculate all of the numbers and give you the new values. Try it. Enter “10” in E3 and watch what happens to the Income.

Section 2.3

77

To help get you used to formulas in Excel and how they work, we will use some of our familiar formulas from last week:

Circle Example Pick a cell where you will enter the radius – say B2. Put “2” in B2 as a starting radius. Then we write the formula for area in a cell next to it – C2. Remember the formula for area of a circle is

� = ��

So, in C2 we write “=PI()*B2^2” π variable for radius exponent in Excel

Then you will notice that the area 12.56637 pops up in C2.

Change the radius to “7” and you will be able to see that the area automatically changes. Nifty, isn’t it? You can change the radius to any number you would like and the area calculation will automatically update. Now, the power of Excel doesn’t stop just there. We can see the areas of a whole bunch of radii at the same time. List out several numbers in the cells beneath the “7” in B2. Now, if you copy the formula from C3 and paste it in C4, C5, C6, etc. you will notice that we can make a whole table of areas. If you label the columns, then others that see your spreadsheet will be able to tell what you did. It should look something like this:

Temperature Conversion Example Make a column of numbers that are temperatures in Fahrenheit starting with cell C10. Then type in the formula that converts Fahrenheit to Celsius in D10: “=5/9*(C10 – 32)”

Circle Radius Area

2 12.56637

4 50.26548

6 113.0973

8 201.0619

10 314.1593

12 452.3893

Section 2.3

78

Copy and paste the formula into the cells next to the list of temperatures. See if it looks something like this:

Fahrenheit Celsius

-40 -40.0

-20 -28.9

0 -17.8

15 -9.4

32 0.0

38 3.3

45 7.2

72 22.2

100 37.8

150 65.6

212 100.0

Section 2.3

79

Section 2.3 Exercises Part A 1. Using the formula for a rectangle and a calculator, fill out the following table:

length width Perimeter Area

5 7

14 3

7.2 18.34

13 2.5

15 17

16 33

281 541.5

2. If the unit for length and width in #1 is inch, what are the units for Perimeter and Area? 3. If the unit for length and width in #1 is centimeter, what are the units for Perimeter and Area? 4. Using the formula for a rectangle and a spreadsheet (Create a new file called Formula Practice), fill out the table in #1 using the formula abilities of the spreadsheet. 5. Using the formula for a circle and a calculator, fill out the following table:

radius Circumference Area

3

12

5.1

17

4

38

114

6. If the unit for radius in #5 is feet, what are the units for Circumference and Area? 7. If the unit for radius in #5 is kilometer, what are the units for Circumference and Area? 8. Using the formula for a circle and a spreadsheet, fill out the table in #5 using the formula abilities of the spreadsheet. 9. Using the formula for a cone and a calculator, fill out the following table:

radius height slant height LSA SA Volume

3 4 5

5 12 13

15 8 17

24 7 25

6 8 10

Assignment 2.3a

80

10. If the unit for radius, height and slant height in #9 is inch, what are the units for Lateral Surface Area, Surface Area, and Volume? 11. If the unit for radius, height and slant height in #9 is centimeter, what are the units for Lateral Surface Area, Surface Area, and Volume? 12. Using the formula for a cone and a spreadsheet, fill out the table in #9 using the formula abilities of the spreadsheet. 13. Open your, “Budget and Expense” spreadsheet. Make sure that all budgets and expenses are updated. Using the “sum” formula, create cells that are the totals of your expenses and incomes. This spreadsheet will be submitted in your portfolio.

Assignment 2.3a

81

Answers: 1. length width Perimeter Area

5 7 24 35 14 3 34 42 7.2 18.34 51.08 132.048 13 2.5 31 32.5 15 17 64 255 16 33 98 528 281 541.5 1645 152,161.5

5. radius Circumference Area

3 18.85 28.27 12 75.40 452.39 5.1 32.04 81.71 17 106.81 907.92 4 25.13 50.27 38 238.76 4,536.46 114 716.28 40,828.14

9. radius height slant height LSA SA Volume

3 4 5 47.12 75.40 37.70 5 12 13 204.20 282.74 314.16 15 8 17 801.11 1507.96 1884.96 24 7 25 1884.96 3694.51 4222.30 6 8 10 188.50 301.59 301.59

2. P – in; A – in2

3. P – cm; A – cm2

4. On Spreadsheet

6. C – ft; A – ft2

7. C – km; A – km2

8. On Spreadsheet

10. LSA – in2; SA – in2; V – in3

11. LSA – cm2; SA – cm2; V – cm3

12. On Spreadsheet

13. In Portfolio

Assignment 2.3a

82

Section 2.3 Exercises Part B 1. Using the formula for a cylinder and a calculator, fill out the following table:

radius height Surface Area Volume

5 7

14 3

7.2 18.34

13 2.5

15 17

16 33

281 541.5

2. If the unit for length and width in #1 is inch, what are the units for Surface Area and Volume? 3. If the unit for length and width in #1 is centimeter, what are the units for Surface Area and Volume? 4. Using the formula for a cylinder and a spreadsheet, fill out the table in #1 using the formula abilities of the spreadsheet. 5. Using the formula for a Sphere and a calculator, fill out the following table:

radius Surface Area Volume

3

12

5.1

17

4

38

114

6. If the unit for radius in #5 is feet, what are the units for Surface Area and Volume? 7. If the unit for radius in #5 is kilometer, what are the units for Surface Area and Volume? 8. Using the formula for a sphere and a spreadsheet, fill out the table in #5 using the formula abilities of the spreadsheet.

83

9. Using a spreadsheet fill out the table for a savings account that has a beginning balance of $150 and grows at 7% with an additional $25 added at the end of each year:

year Beginning Balance Ending Balance

1 150 150 × 1.07 + 25 = 185.5

2 185.5 185 × 1.07 + 25 =

. . .

use your calculator to make sure that the spreadsheet is calculating it correctly.

15

10. As a group, select a typical job that one of you anticipates having in the next five years. Then open a spreadsheet document and go through the lifelong income example in this section. How much money do you expect to earn over your lifetime?

Assignment 2.3b

84

Answers: 1. radius height Surface Area Volume

5 7 376.99 549.78 14 3 1,495.40 1,847.26 7.2 18.34 1,155.40 2,986.86 13 2.5 1,266.06 1,327.32 15 17 3,015.93 12,016.59 16 33 4,926.02 26,540.17 281 541.5 1,452,185.50 134,326,275.61

5. radius Surface Area Volume

3 113.10 113.10 12 1,809.56 7,238.23 5.1 326.85 555.65 17 3,631.68 20,579.53 4 201.06 268.08 38 18,145.84 229,847.30 114 163,312.55 6,205,877.00

2. SA – in2; V – in3

3. SA – cm2; V – cm3

4. On Spreadsheet

6. SA – ft2; V – ft3

7. SA – km2; V – km3

8. On Spreadsheet

9. At the end of 15 years you should have $1,042.08

10. Complete when everyone can do it on their own.

Assignment 2.3b

85

Section 2.3 Exercises Part C Chapter 2 Exam Review

Find the following:

1. )5(83673−⋅+ 2. mv ⋅⋅⋅ 92 3. )47(4)73(6 2

+⋅−−

Find the perimeter of the following shapes:

4. 12 11 s-4 t+3 8

5. 18 13 f+2

6. 18 7 8 21 r-9

Find the following when p = -5, r = 7, t = 32 , and a = 4.

7. 12 – a3 8.

412 - 7a 9. 2r – 3p + 9t

Use the formula for distance, rate and time.

10. r = 6 m/h t = 19 hours d =

11. r = 65 km/h t = 4.3 hours d =

12. r = 36 feet per second

t = 312 seconds

d =

Use the formula for Interest.

13. P = $2,800 r = 7% t = 4 A =

14. P = $5,000 r = 6% t = 9 A =

15. P = $300 r = 13% t = 7 A =


16. q = 15 d = 27 V =

17. p = 30 d = 25 V =

18. p = 37 q = 23 n = 7 V =

Assignment 2.3c

86

Use the formula for Temperature Conversion.

19. F = 212 C =

20. F = 98.6 C =

21. F = -40 C =

Use the formulas for a cone.

22. r = 6 m h= 7 m V =

23. r = 9 ft l = 12.8 ft SA =

24. r = 3 in l = 7.9 in LSA =

Use the formulas for a rectangle.

25. l = 3 yd w = 5 yd A =

26. l = 10.7 cm w = 4 cm A =

27. l = 8.6 mm w = 9 mm P =

Use the formulas for a circle.

28. r = 4 in C =

29. r = 15 in A =

30. r = 7 m C =

Use the formulas for rectangular solid.

31. l = 7 cm w = 2 cm h = 8 cm SA =

32. l = 4.2 mi w = 5 mi h = 7mi V =

33. l = 6 km w = 8 km

h = 2�

� km

SA =

34. Create a Visual Chart on one side of a piece of paper for Chapter 2 material including information and examples relating to Calculator and Spreadsheet Usage and Formulas.

Assignment 2.3c

87

Answers:

1. 321 31. 172 cm2

2. 18vm 32. 147 mi3

3. -200 33. 166 km2

4. 30 + s + t

5. f + 33

6. r + 45

7. -52

8. 4

9. 35

10. 114

11. 279.5

12. 84

13. $3,670.23

14. $8,447.39

15. $705.78

16. $6.45

17. $2.80

18. $6.47

19. 100° C

20. 37° C

21. -40° C

22. 263.89 m3

23. 616.38 ft2

24. 74.46 in2

25. 15 yd2

26. 42.8 cm2

27. 35.2 mm

28. 25.13 in

29. 706.86 in2

30. 43.98 m

Assignment 2.3c

88

Chapter 3

ALGEBRA

Overview

Algebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents

89

3+ “what” = 7? If you have come through arithmetic, the answer is fairly obvious: 4. However, if I were to ask something like: 2 times “what” plus 5 all divided by 7, then minus 6 = 5? There tends

to be a little more difficulty in popping out the answer. The beauty of math is that it allows us to write down all of that stuff and then systematically make it simpler and simpler until we have only the number left. Wonderful. We start with the easy ones to find out all of the rules and then we will build up to the big ones. 3+ “what” = 7 First, we need to adjust the fact that we are going to be writing “what” all the time. A very common thing is to put a letter in that place that could represent any number. We call that a variable. We replace the word “what” with “x” (or you could use p, q, r, f, m, l . . . ) So our equation becomes: 3 + x = 7 The whole goal of math is to find the number that makes that statement true. We already know that the number is 4. We would write: x = 4 Now, look at what happened to our original equation. Do you see that the right side is missing a 3 and the left side is now 3 lower as well. This gives us some insight into what we can do to equations! Try another one: x + 8 = 10 What number would make that statement true? If x were equal to 2, it would work. We write: x = 2 Notice how we get the number that would work by subtracting that 8 from both sides of the equation. Let’s see if it works with some other equations: x – 7 = 2 x – 3 = 10 With these two equations, the answers are: x = 9 and x = 13 We got the answers by adding the 7 and the 3 to the right hand sides. This brings up a good point. In the first couple of equations that we did, we subtracted when the equation was adding. In the next two equations, we added when the equation was using subtraction. Let’s look at what happens when we start doing multiplication: 4x = 20. What number would work? That is right, 5.

Section 3.1 Linear Equations

Section 3.1

90

A great way to think about these concepts is as though you have a balance that is centered on the equal sign. As long as you put the same thing on both sides, you remain balanced.

x = 5 What would you do to 20 to get 5? Divide by 4. Holy smokes! That is the exact opposite of what the equation is doing. Here is another:

7x = 4

What number divided by 7 equals 4? That’s it, 28. We times 4 by 7 to get that answer. Multiplying by 7 is the exact opposite of dividing by 7. This leads us to a couple of conclusions that form the basis for everything we will do in Algebra:

1) When we want to get rid of numbers that are surrounding the variable, we need to do the opposite (technically called the inverse) of them.

2) We can add, subtract, multiply, or divide both sides of an equation by any number and still have the equation work. Here is how it would work, one of each:

x+7=11 4x=24 x-3=24 5x =7

-7 -7 /4 /4 +3 +3 (5) (5) x = 4 x = 6 x = 27 x = 35 You may ask why we go through all of that when the answers are obvious. The answer is that these problems will not be so easy later on, and we need to practice these easy ones so that when we get the hard ones, they crumble before our abilities. Now to some which are a little tougher. When we have one like this: 2x – 7 = 11 We could think about it long enough to find a number that works, and maybe you can do that, but I have to tell you that in just a little while we are going to have a problem that you won’t be able to do that with too quickly. So, let’s use what we learned to get rid of the 2 and the 7 so that x will be left by itself. If you remember the order of operations, you will remember that the 2 and the x are stuck together by multiplication, so we can’t get rid of the 2 until the 7 has been taken care of like this: 2x – 7 = 11 2x = 18 (we added 7 to both sides) x = 9 (divided both sides by 2)

Section 3.1

91

To illustrate the idea of un-doing operations, I would like to try to stump you with math tricks. We begin. I am thinking of a number, and it is your job to guess what the number is.

I am thinking of a number. I times the number by two. I get 10.

Not too hard to figure out, you say? You're right. The answer is 5 and you obtained that by taking the result and going backwards. Try the next one:

I am thinking of a number. I times the number by 3. Then I subtract 5. Then I divide that number by 2. Then I add 4 to that. I get 18. What was the number I started with?

Aha. A little tougher don't you think? Well, If you think about it just one step at a time, then the thing falls apart. What number would I add 4 to to get 18? 14 (notice that it is just 18 subtract 4). We can just follow up the line doing the exact opposite of what I did to my number. Here you go:

Start with 18 Subtract 4 = 14 Multiply by 2 = 28 Add 5 = 33 Divide by 3 = 11.

That's it! Most of Algebra is summed up in the concept of un-doing what was done.

I am thinking of a number. I times it by 4. Then I add 5. Then I divide by 9. Then I subtract 7. I get -2. What did I start with?

This one is done the same way as the other one but I wanted to show you how you make that into an equation that will be useful in the rest of your math career. Instead of writing each

Section 3.1

92

step out, we construct an equation. We write it again but this time we will write the equation along with it:

I am thinking of a number. We call that x.

I times it by 4. 4x

Then I add 5. 4x+5

Then I divide by 9. 9

54 +x

Then I subtract 7. 79

54 −+x

I get -2. What did I start with? 79

54 −+x = -2

That looks like a nasty equation, but it is done in exactly the same way. We just go backwards and un-do all of the things that were done to the original number. We are using the rule that we can add, subtract, multiply or divide both sides of the equation by the same thing. I know you can do it when it is all written out, so I will show you what it looks like using the equation:

79

54 −+x = -2

9

54 +x = 5 add 7 to both sides

4x + 5 = 45 times both sides by 9 4x = 40 subtract 5 from both sides x = 10 divide both sides by 4.

10 is the number I started with! Go ahead and make sure by sticking it into the original problem, and you will see that we found the right number. We call that number a solution, because it is the only number that solves the equation.

Solving for a variable: When given a formula, it is sometimes requested that you solve that formula for a specific variable. That simply means that you are to get that variable by itself. An example: Solve for t: rt = d (Original equation of rate x time = distance) We are supposed to get t by itself. How do we get rid of the “r”? Divide both sides by r. It looks like this rt = d

r

rt=

r

d

t = r

d Done. t is by itself.

Notice here that we are still undoing in the opposite order of what was there.

Section 3.1

93

Another example: Solve for x: y = bx +c y – c = bx subtract “c” from both sides

xb

cy=

− Divide both sides by “b”.

Done. “x” is by itself.

Section 3.1

94

Section 3.1 Exercises Part A Find the Volume of a rectangular solid when the width, height and length are given. Formula is V=lwh

1. l = 4 in w = 2.5 in h = 3 in V =

2. l = 7 ft w = 4 ft h = 2.8 ft V =

3. l = 7.2 m w = 9 m h = 3 m V =

Find the Area of a trapezoid when the bases and height are given. Formula is

A = 21 h(B+b)

4. B = 15 b = 10 h = 7 A=

5. B = 21 b = 11 h = 3 A=

6. B = 19 b = 6 h = 10 A=

Simplify.

7. 2(3+x)+5(x-7) 8. 5(a-3b) – 4(a-5) 9. 3x+4y-7z+7y-3x+18z

10. 2s(t-7) – 6t(s+3) 11. 3(x2-5n) +3n – 7x2 12. 6kj – 7k +8kj +11

Solve.

13. 352

7

135 =

+

−x

14. 1237

6

823 =−

+

−− x

15. -3 + m = 18

16. 37 t = 14 17. -13 = 5x + 7 18.

34

65=

−x

19. 83− x – 4 = 20 20. 12 + 2p = 3 21. .4y = 78

Example:

4x + x – 7 = 1

Combine x’s

5x – 7 = 1

+7 on both sides

5x = 8

x = 58

Divide by 5 on both sides

Assignment 3.1a

95

22. 5x + 3 – 7x = 15 23. 3x – 9 + 2x = - 3 24. .3p + 5 = 19

25. -r + 9 = -15 26. 4f + 9 = 9 27. 11

5

32=

+x

28. t + t + 4t – 7 = 17 29. 1837

6

853 =−

+

−x

Solve for the specified variable.

30. y = mx + b for b 31. r

m=

−

3

75 for m

32. A = 2πrh for h 33. A = 21 bh for b

34. C = 95 (F – 32) for F 35. V =

31 πr2h for h

Preparation. 36. After reading some from the next section, Try to solve this problem. Two numbers add up to 94 and the first is 26 more than the second one. Find the two numbers. 37. Find the missing variable for a cone: r = 9 l = SA = 622.04

Assignment 3.1a

96

Answers:

1. 30 in3 29. x =

58 or 1.6

2. 78.4 ft3 30. b = y – mx

3. 194.4 m3 31.

m = 5

73 +r

4. 87.5 32. r

Ah

π2=

5. 48 33. h

Ab

2=

6. 125 34. F = 59 C + 32

7. 7x – 29 35. 2

3

r

Vh

π=

8. a – 15b + 20

9. 11y + 11z

10. -4st – 14s – 18t

11. -4x2 - 12n

12. 14kj – 7k + 11

13. x = 12

14. x = 2

15. m = 21

16. t = 6

17. x = -4

18. x = 5

18 or 3.6

19. x = -64

20. p = -29

21. y = 195

22. x = -6

23. x = 56 or 1.2

24. p = 46.6

25. r = 24

26. f = 0

27. x = 26

28. t = 4

Assignment 3.1a

97

Applications of linear equations “When am I ever going to use this?” “Where would this be applicable?” All the way through math, students ask questions like these. Well, to the relief of some and the dismay of others, you have now reached the point where you will be able to do some problems that have been made out of real life situations. Most commonly, these are called, “story problems”. The four main points to remember are:

D- Data. Write down all the numbers that may be helpful. Also, note any

other clues that may help you unravel the problem.

V- Variable. In all of these story problems, there is something that you

don’t know, that you would like to. Pick any letter of the alphabet to represent this.

P- Plan. Story problems follow patterns. Knowing what kind of problem

it is, helps you write down the equation. This section of the book is divided up so as to explain most of the different kinds of patterns.

E- Equation. Once you know how the data and variable fit together.

Write an equation of what you know. Then solve it. This turns out to be the easy part.

Once you have mastered the techniques in solving linear equations, then the fun begins. Linear equations are found throughout mathematics and the real world. Here is a small outline of some applications of linear equations. You will be able to solve any of these problems by the same methods that you have just mastered.

Translation The first application is when you simply translate from English into math. For example: Seven less than 3 times what number is 39? Since we don’t know what the number is, we pick a letter to represent it (you can pick what you would like to); I will pick the letter x: 3x – 7 = 39 then solve 3x = 46

x =346 (or

3115 or 15.3)

That’s the number.

Substitution Sometimes you are given a couple of different things to find. Example: Two numbers add to 15, and the second is 7 bigger than the first. What are the two numbers?

Section 3.1 Applications: Translation

98

Pick some letters to represent what you don’t know. Pick whatever is best for you. I will choose the letter “f” for the first number and “s” for the second. I then have two equations to work with: f + s = 15 and s = f + 7 f + f + 7 = 15 2f + 7 = 15 2f = 8 f = 4 4 must be the first number, but we need to stick it back in to one of the original equations to find out what “s” is. s = f + 7 = 4 + 7 = 11. 4 and 11 are our two numbers. These kind of problems often take the form of an object being cut into two pieces. Here, I will show you what I mean. Example: A man cuts a 65 inch board so that one piece is four times bigger than the other. What are the lengths of the two pieces? Now, I would personally pick “f” for first and “s” for second. We know that f + s = 65 and that s = 4f Thus, f + 4f = 65 5f = 65 f = 13, so the other piece must be 52. The pieces are 13in and 52in.

Shapes With many of the problems that you will have, pictures and shapes will play a very important role. When you encounter problems that use rectangles, triangles, circles or any other shape, I would suggest a few things:

1. Read the problem 2. READ the problem again. 3. READ THE PROBLEM one more time.

Once you draw a picture to model the problem – read the problem again to make sure that your picture fits. The formulas for the shapes that we will be discussion are found in Section 2.2.

The letter “s” and “f+7” are exactly the same and can be changed places.

Sections 3.1 Applications: Substitution/ Shapes

99

Variable on Both Sides Unfortunately, not all equations come out such that this un-doing technique works. Sometimes the x shows up in several different places at once: 3x – 5 +2x – 3 = 5x + 7(x – 8) Seeing all of the x’s scattered throughout the equation sometimes looks daunting, but it isn’t as bad as all that. We know a couple of ways to make it look a bit more simple. 3x – 5 +2x – 3 = 4x + 7(x – 8) becomes 5x – 8 = 4x + 7x – 56 Distribute the 7 and combine 5x – 8 = 11x – 56 Combine the like terms Now we reach a point where you should feel somewhat powerful. Remember that you can add, subtract, multiply or divide anything you want! (As long as you do it to both sides). Particularly, I don’t like the way that 11x is on the left hand side. I choose to get rid of it! So, I subtract 11x from both sides of the equation: 5x – 8 = 11x – 56 -11x -11x Upon combining the like-terms, I get -6x – 8 = -56 Which now is able to be un-done easily: -6x = -48 (add 8 to both sides) x = 8 (divide both sides by -6)

Special cases: What about 2x + 1 = 2x + 1

Well if we want to get the x’s together we had better get rid of the 2x on one side. So we subtract 2x from both sides like this: 2x + 1 = 2x + 1 -2x -2x

You might as well know that if you didn’t like the 5x on the right hand side, you could get rid of that instead: 5x – 8 = 11x – 56 -5x -5x Combining like terms, we get: -8 = 6x – 56 48 = 6x (add 56 to both sides) 8 = x (divide both sides by 6) We will always get the same answer!

You can’t mess up!

Section 3.1

100

1 = 1 Ahh! The x’s all vanished. Well, what do you think about that? This statement is always true no matter what x is. That is the point. x can be any number it wants to be and the statement will be true. All numbers are solutions.

On the other hand try to solve: 2x + 1 = 2x - 5

-2x -2x 1 = -5 Again, the x’s all vanished. This time it left an equation that is never true. No matter what x we stick in, we will never get 1 to equal -5. It simply will never work. No solution.

Section 3.1

0 = 0 5 = 5 -3 = -3 solution is all real numbers

0 = 1 5 = 7 -3 = 2 No solution

101

Section 3.1 Exercises Part B Simplify.

1. 4s(t-9) –t(s+11) 2. 12(x2-5n) +3n – 4x2 3. 6nj – 7j +8nj +11n

Solve.

4. 302

5

465 =

+

−x

5. 1225

6

827 =−

+

+− x

6. -3 – 7m = 18

7. 27 t = -14 8. -15 = 3x + 9 9.

333

72=

−x

10. t +5t + 4t – 7 = 17 11. 4237

6

859 =−

+

−x


12. y = mx + b for x 13. r

m=

+

2

95 for m

14. 6 = 7b – pb for b 15. 3t + nt = y for t

16. P = 2l + 2w for l 17. SA = 2πrh+2πr2 for h

18. 27 is 6 more than 3 times a number. What is the number? 19. 18 less than 5 times a number is 52. What is the number? 20. Two numbers add to 37 and the second is 9 bigger than the first. What are the two numbers? 21. Two numbers add to 238 and the first is 34 bigger than the second. What are the two numbers? 15in 22. Find the area of the shaded region: 8in

Assignment 3.1b

102

23. I have created a triangular garden such that the largest side is 6ft less than twice the smallest and the medium side is 5ft larger than the smallest side. If the total perimeter of the garden is 47ft, what are the lengths of the three sides? 24. If a rectangle’s length is 5 more than twice the width and the perimeter is 46 mm, what are the dimensions or the rectangle? 25. If a cone has a Lateral Surface Area of 250 in2, a radius of 8in, what is the slant height of the cone? Use a calculator. 26. Two numbers add to 589 and the first is 193 bigger than the second. What are the two numbers? 27. If a cylinder has a volume of 538 cm3 and a radius of 6 cm, how tall is it? 28. Find the missing variable for a rectangle: P = 39 ft w = 7.2 ft l = 29. Find the missing variable for a cylinder: SA = 800 in2

h = r = 9 in

Solve.

30. 4p + 2 = 7p - 6 31. -4n + 5 = n 32. 2x – 7 = x + 5

33. x – 42 = 15x 34. – 4(x-3) = -2x +12 35. 7x = 13 + 7x

36. .4x – 1 = .9x + 5 37. 2(x – 4) = 3x - 14 38. .4y +78 = 78 + .4y

Example:

x + 4 – 5x = 7x + 1 Combine like terms

-4x + 4 = 7x + 1 +4x +4x

Get all x’s together by adding 4x to both sides

4 = 11x + 1 -1 -1

Subtract 1 from both sides

3 = 11x

113 = x Divide both sides by 11

Assignment 3.1b

103

Answers:

1. 3st – 36s – 11t 28. l = 12.3 ft

2. 8x2 – 57n 29. h = 5.15 in

3. 14nj – 7j + 11n 30. p = �

� or 2.67

4. x = 4 31. n = 1

5. x = 13 32. x = 12

6. m = -3 33. x = -3

7. t = -4 34. x = 0

8. x = -8 35. No solution

9. x = 53 36. x = - 12

10. t = 5

12 or 2.4 37. x = 6

11. x = -54 38. All numbers

12. x = m

by −

13. 5

92 −=

rm

14. b =

p−7

6

15. t = n

y

+3

16. l =

2

2wp −

17. h =

r

rSA

π

π

2

2 2−

18. 7

19. 14

20. 14, 23

21. 102, 136

22. 69.73 in2

23. 12, 17, 18

24. l = 17mm; w = 6mm

25. ℓ = 9.95 in

26. 198, 391

27. 4.76 cm

Assignment 3.1b

104

Section 3.1 Exercises Part C Solve.

1. 652

5

435 =

+

+x

2. 20173

5

823 =+

−

+− x

3. -17 – 7m = -18

4. 73 t + 1 = -11 5. 9 = 3x +17 6.

134

75=

+x

7. 8t +3t + 14t – 17 = -17 8. 1839

2

857 =−

+

+x


15. 48 is 9 more than 3 times a number. What is the number? 16. 18 less than 7 times a number is 80. What is the number? 17. Two numbers add to 151 and the second is 21 bigger than the first. What are the two numbers? 18. Two numbers add to 436 and the first is 134 bigger than the second. What are the two numbers? 19. Find the area of the shaded region: 3cm 8cm 14cm 20. I have created a triangular garden such that the largest side is 9 less than twice the smallest and the medium side is 7 larger than the smallest side. If the total perimeter of the garden is 82, what are the lengths of the three sides?

9. p= fx + bn for f 10.

2

xzxfF

−= for f

11. M = 5t – 3p for t 12. LSA = πrl for r

13. E =

2

1

T

TQ − for Q

14. c

gs=

−

7

43 for g

Assignment 3.1c

105

21. If a rectangle’s length is 7 more than 4 times the width and the perimeter is 54 what are the dimensions or the rectangle? 22. If a cone has a volume of 338 cm3 and a radius of 6 cm, how tall is it? 23. Find the missing variable for a parallelogram: A = 64 in2

h = b = 12.6 in Solve.

24. 5p + 12 = 33 – p 25. 7n + 18 = 5(n – 2) 26. 5x – 10 = 5x + 7

27. x – 7 = 15x 28. 2x – 4(x-3) = -2x +12 29. .07x = 13 - .12x

30. .7(3x – 2) = 3.5x + 1 31. .3x – 9 + 2x = 4x - 3 32. .4y = 78 + .4y

33. 7(x – 5) – 3x = 4x – 35 34. 9x – 4(x – 3) = 15x 35. 2x – 3x + 7x = 9x +8x

Example:

x + 4 – 5x = 7x + 1 Combine like terms

-4x + 4 = 7x + 1 +4x +4x

Get all x’s together by adding 4x to both sides

4 = 11x + 1 -1 -1

Subtract 1 from both sides

3 = 11x

113 = x Divide both sides by 11

Assignment 3.1c

106

Answers:

1. x = 17 28. All real numbers

2. x = -6 29. 68.42

3. m = 71 30. x = -

712 or -1.71

4. t = -28 31. x = -3.53

5. x = -38 32. No solution

6. x = 9 33. All numbers

7. t = 0 34. x = 56 or 1.2

8. x = -4 35. x = 0

9. f =

x

bnp −

10. f =

x

xzF +2

11. t =

5

3pM +

12. l

LSAr

π=

13.

2

1

T

TEQ +=

14. 4

37

−

−=

scg or

4

73 cs −

15. x = 13

16. x = 14

17. 65, 86

18. 151, 285

19. 129.9 cm2

20. 21, 28, 33

21. w = 4, l = 23

22. h = 8.97 cm

23. h = 5.08 in

24. p = 27 or 3.5

25. n = -14

26. no solution

27. x = -21

Assignment 3.1c

107

Percents If you scored 18 out of 25 points on a test, how well did you do. Simple division tells us that you got 72%. As a review, 18/25 = .72 If we break up the word “percent” we get “per” which means divide

and “cent” which means 100. Notice that .72 is really the fraction 10072 . We

see that when we write is as a percent instead of its numerical value, we move the decimal 2 places. Here are some more examples to make sure that we get percents: .73 = 73% .2 = 20% .05 = 5 % 1 = 100% 2.3 = 230% The next reminder, before we start doing problems, is that the word “of” often means “times”. It will be especially true as we do examples like: What is 52% of 1358? All we need to do is multiply (.52)(1358) which is 706.16 Sometimes however, it isn’t quite that easy to see what needs to be done. Here are three examples that look similar but are done very differently. Remember “what” means “x”, “is” means “=” and “of” means times. 15 is what percent of 243? 15 is 243% of what? 15 = x (243) 15 = 2.43x .062 = x 6.17 = x 6.2% = x Once we have that down, we have the ability to solve tons of problems involving sales tax, mark-ups, and discounts. Here are two examples:

An item sold at $530 has already been marked up 20%. What was the price before the mark-up? x + .2x = 530 original + 20% of original = final price 1.2x = 530 x = 441.67

What is 15% of 243? x = .15(243)

x = 36.45

An item sells for $85.59 but is on sale at 20% off. What is the final price? .2(85.59) = 17.12 amount of discount 85.59 – 17.12 subtract discount

$68.47 = final price

Section 3.2 Applications: Percents

Section 3.2 Linear Equations w/ fractions

108

Section 3.2 Exercises Part A 1. 45 is 12 more than 3 times a number. What is the number? 2. 25 less than 7 times a number is 108. What is the number? 3. Two numbers add to 251 and the second is 41 bigger than the first. What are the two numbers? 4. Two numbers add to 336 and the first is 124 bigger than the second. What are the two numbers? 5. Find the area of the shaded region: 5cm 8cm 15cm 6. I have created a triangular garden such that the largest side is 8m less than twice the smallest and the medium side is 12m larger than the smallest side. If the total perimeter of the garden is 104m, what are the lengths of the three sides? 7. If a rectangle’s length is 5 more than 3 times the width and the perimeter is 58 mm what are the dimensions of the rectangle? 8. If a parallelogram has an area of 258.9 cm2 and a base of 23.2 cm, how tall is it? 9. Find the missing variable for a trapezoid: A = 68 ft2

b = h= 4ft B = 21ft Solve.

10. 7p + 13 = 33 – 4p 11. 5n + 48 = 7n – 2(n – 2) 12. 5x – 10 = 7(x – 2)

13. 3x – 7 = 12x 14. 5x – 7(x+3) = -2x -21 15. .06x = 15 - .18x

16. .8(7m – 2) = 9.5m + 1 17. .2q – 7 + 2q = 3q - 5 18. 12t = 45 + .4t

Assignment 3.2a

109

19. 6(x – 5) – x = 5x – 20 20. 9x – 2(x – 3) = 15x +7 21. 5x – 13x + x = 7x +8x

22. 18 is what percent of 58? 23. What is 87% of 54?

24. 34 is 56% of what? 25. What is 13% of 79?

26. 119 is 8% of what? 27. 23 is what percent of 74?

28. Original Price:$92.56 Tax: 7.3% Final Price:

29. Original Price: Discount: 40% Final Price: $43.90

30. Original Price: Tax: 5% Final Price: $237.50

31. Original Price: $58.50 Discount: 30% Final Price:

32. If the population of a town grew 21% up to 15,049. What was the population last year? 33. If the price of an object dropped 25% down to $101.25, what was the original price? Preparation. 34. After reading some from the next section, try to solve this equation.

7

27

157

137

xx −=+

35. Solve.

3

23

153

133

xx −=+

Assignment 3.2a

110

Answers:

1. x = 11 28. $99.32

2. x = 19 29. $73.17

3. 105, 146 30. $226.19

4. 106, 230 31. $40.95

5. 136.71 cm2 32. 12,437

6. 25m, 37m, 42m 33. $135

7. w = 6mm, l = 23mm 34. Discuss together.

8. 11.16 cm 35. Discuss together.

9. 13ft = b

10. p = 1120

11. No solution

12. x = 2

13. x = 97−

14. All numbers

15. x = 62.5

16. m = - 32 or -

3926

17. q = -2.5

18. t = 3.879

19. No solution

20. x = -81

21. x = 0

22. 31%

23. 46.98

24. 60.7

25. 10.27

26. 1487.5

27. 31%

Assignment 3.2a

111

Equations with Fractions The one other thing that might throw you off is when you see a bunch of fractions in the problem. Not to worry, remember that you have power to do anything you want to the equation. For example:

8

7

8

5

8

3 xx =− might be easier to look at if there weren’t so many fractions in the

way. Well, get rid of them. Multiply by 8 on both sides.

8

7

8

5

8

3 )8()8()8( xx =− which makes it become:

3x – 5 = 7x (not bad at all) -5 = 4x

-45 = x Ta Da.

Worse example:

54

3

7

2=

−−

x looks scary.

You have the ability to wipe out all of the fractions. Fractions are simply statements of division. The opposite of division is multiplication – and you have the power to multiply both sides of the equation by anything you want to. The question is, what will undo a division by 7 and by 4; the answer is multiplication by 28. Here is what it looks like:

54

3

7

2=

−−

x

)28(54

3)28(

7

2)28( =

−−

x (multiplying everything by 28)

(4)2 – (7)(x – 3) = 140 (28/7 = 4 and 28/4 = 7) 8 – 7x + 21 = 140 (Distribute the -7) -7x +29 = 140 (Combine numbers) -7x = 111 (Subtract 29 from both sides)

x = 7

111− (Not a nice looking answer, but it is

right!)

Section 3.2

1. Simplify

2. Subtract 3. Divide

112

Every problem can be boiled down to three steps:

Section 3.2

Linear Equations

1. Simplify

2. Add/Subtract

3. Multiply/Divide

1. Parentheses 2. Fractions 3. Combine like terms

113

Section 3.2 Exercises Part B 1. 35 less than 7 times a number is 98. What is the number? 2. Two numbers add to 351 and the second is 71 bigger than the first. What are the two numbers? Solve.

3. 7p + 12 = 33 – 4p 4. 3n + 48 = 7 – 2(n – 2) 5. 5x – 10 = 5(x – 2)

6. 3x – 7 = 15x 7. 5x – 7(x+3) = -2x +12 8. .09x = 13 - .18x

9. .8(3x – 2) = 9.5x + 1 10. .2x – 7 + 2x = 3x - 5 11. 12m = 70 + .4m

12. 5(x – 5) – x = 4x – 20 13. 9x – 4(x – 3) = 15x +7 14. 8x – 12x + x = 9x +8x

15. 85 is what percent of 39? 16. 85 is 54% of what?

17. What is 19% of 2,340? 18. What is 23% of 79?






25. If the population of a town grew 31% up to 17,049. What was the population last year? 26. If the price of an object dropped 35% down to $101.25, what was the original price?

Assignment 3.2b

114

Solve.

27. 37 t – 5 = 19 28.

83− (x – 7) = 5 + 3x 29.

32 x – 6 = 3 +

21 x

30. 54 x = 2x -

35 31.

53 x –

52 (x-3) =

51 x +3 32.

514

723 −+ = xx

33. .9(-4x – 5) = 2.5x + 6 34. .0005x + .0045 = .004x 35. 65

47 8 −=+x x

Preparation. 36. Describe the best way to get rid of fractions in an equation.

Example:

�

�� + 4� −

=

�

�� +

(12)�

�� + 4� −

(12)= (12)�

�x +(12)

Clear fractions by multiplying by 12

4(x+4) – 30 = 3x + 10

4x + 16 – 30 = 3x + 10 Distribute through parentheses

x – 14 = 10 Combine, getting x to one side

x = 24 Add 14 to both sides

Assignment 3.2b

115

Answers:

1. 19 28. x = -2719

2. 140, 211 29. x = 54

3. p = 1121 30. x =

1825

4. n = 5

37− or -7.4 31. no solution

5. All numbers 32. x = 1317

6. x = 127− 33. x = -

61105

7. no solution 34. x = 79

8. x = 48.15 35. x = 1375

9. x = -.366 36. Discuss together.

10. x = -2.5

11. m = 6.03

12. no solution

13. x = 21

14. x = 0

15. 218%

16. 157.4

17. 444.6

18. 18.17

19. 661.1

20. 24.7%

21. $77.86

22. $71.00

23. $323.33

24. $33.30

25. 13,015

26. $155.77

27. t = 772

Assignment 3.2b

116

Summary of Linear Equations

Linear Equations

1 – Simplify 2 – Add/Subtract 3 – Multiply/Divide

Word Problems D,V,P,E

Parentheses Fractions Combine like terms

Section 3.2

117


Solve. 1.

7025

135 =

−

−x

2. 1953

2

463 =−

+

+− x

3. -4 – 9m = -22

4. 76 t = -24 5. 19 = 3x -7 6.

93

75−=

−x


7. V

t

ats=

−

2

2 2

for s 8.

r = pt

I for p

9.

12

2

RR

LRd

+= for R1

10. c

gs=

−

11

59 for s

11. 84 is 6 more than 3 times a number. What is the number? 12. Two numbers add to 438 and the first is 74 bigger than the second. What are the two numbers? 14in 13. Find the area of the shaded region: 9in 14. If a rectangle’s length is 7 more than 4 times the width and the perimeter is 194 mm, what are the dimensions or the rectangle? 15. Find the missing variable for a rectangle: P = 48.3 ft w = 7.2 ft l = 16. Find the missing variable for a cone: SA = 628.32 in2

r = 8 in l = Solve.

17. 7p + 12 = 13 – 7p 18. 4n + 68 = 7 – 2(n – 2) 19. 7x – 10 = 5(x – 2)

Assignment 3.2c

118

20. 9x – 4 = 15x 21. 8x – 7(x+3) = x – 21 22. .18x = 13 - .20x

23. 14 is what percent of 68? 24. What is 37% of 754?




29. If the price of a meal after a 20% tip was $28.80? What was the price of the meal before the tip was added? 30. If the price of an object dropped 15% down to $59.50, what was the original price? Solve.

31. 37 t – 2 = 19 + 5t 32.

43− (x – 4) = 5 + 2x 33.

61 x – 4 = 3 +

103 x

34. 25 (-4x – 2) =

43 x + 6 35.

685

35 +− = xx 36.

73

147 6 −=+x x

Assignment 3.2c

119

Answers:

1. x = 27 28. $67.54

2. x = -1 29. $24

3. m = 2 30. $70

4. t = -28 31. t = -863

5. x = 326 32. x = -

118

6. x = -4 33. x = -52.5

7.

2

2 2atVt

s+

= 34. x = -

4344

8. rt

Ip =

35. x = -6

9. d

dRLRR

221

−=

36. x = 11

10. 9

511 gcs

+=

11. 26

12. 182, 256

13. 62.38 in2

14. 18mm X 79mm

15. l = 16.95 ft

16. 17 in

17. p = 141

18. n = -9.5

19. x = 0

20. x = -32

21. All numbers

22. x = 34.21

23. 20.6%

24. 278.98

25. 661.1

26. 36.5%

27. $206.62

Assignment 3.2c

120

The rules that come with exponents are relatively easy to understand, but they take some practice to ensure that you have them down completely. Instead of numbers we will use letters. If we multiply:

x5x8 We just have to remember what that means: (xxxxx)(xxxxxxxx), which is simply 13 of them multiplied together. We write it as x13.This is our very first rule! Exponents add during multiplication. x5x8=x13. The next one is quite similar: (x5)8 Again, we just have to remember what it means: (x5)(x5)(x5)(x5)(x5)(x5)(x5)(x5) which is by the first rule: x40. That gives us the second rule: Exponent to exponent will multiply. (x5)8 = x40. Division with exponents is just about as easy. Looking at:

��

This means: �� and we are left with x3.

Third rule: Exponents subtract during division. This particular rule gives rise to a couple of interesting facts. Specifically, what happens if the top and the bottom have the same power?

�� = x0

But, we know that anything divided by itself is equal to 1. Thus: x0 = 1 Secondly, what happens if the number on the bottom is larger than the one on the top. For example:

��

By using the third rule we get

�� = x�� =x-3, - a negative exponent! What do we do with that?

Well, if we do it the long way, we get:

�� which is

��

Section 3.3 Exponents

Section 3.3

121

Thus we have our next definition. A negative exponent puts the number on the bottom.

x-3 = ��

Look at a couple of examples: Using rules of exponents Checking with numbers 23

· 22 = 25 = 2 · 2 · 2 · 2 · 2 = 32 23 · 22 = 8 · 4 = 32

� � = 2�� = 2� = 8

� � = � �� = 8

Look at that. The rules really work for any number. Here are some more examples to be able to simplify some expressions: (3x5)3 = 27x15 by use of the second rule. (4y5)(7y12) = 28y17 by use of the first rule.

2-6 = � � = �� by the definition of a negative exponent

6-2 = �� = ��

�� = �� = 49

�� = �� = �� = �� = �� = �� Here is a summary of how you can simplify expressions with exponents:

Rule Official Example Why Multiplication – add exponents

aman = am+n 3x2 · 2x5 = 6x7 3xx2xxxxx = 6xxxxxxx = 6x7

Exponent to a power – multiply exponents

(am)n = amn (5x2)3 = 125x6 (5x2) (5x2) (5x2) = 125x6

Division – subtract exponents

a a! = a ! 36x�4x� = 9x 36xxxxxxxx4xxxxxx = 9xx = 9x Exponent of 0

a0 = 1 if a ≠ 0 70 = 1; x0 = 1 1= �� = x$ by division rule

Negative exponent

a-n = �%&

2-4 = � � = �� ; 1x� =x�

1x� = xxxxxxxxx = x x� = x�

Section 3.3

122

Section 3.3 Exercises Part A Simplify the following.

1. (3m2)5 2. x7x11

3. (��(� 4. t8t5

5. 3-4 6. 3x7 · 4x

7. �)�� )� 8. 170

9. (g8)-2 10. ��*� �*�

11. (2m2n5g8)7 12. 5x2 · 4x7

13. 25 less than 7 times a number is 73. What is the number? 14. Two numbers add to 251 and the second is 41 bigger than the first. What are the two numbers? Solve.

15. 5p + 12 = 39 – 4p 16. 5n + 48 = 7n – 2(n – 2) 17. 15x – 10 = 5(x – 2)

18. 2(x – 5) – x = 4x – 7 19. 9m – 3(m – 3) = 15m +7 20. 8x – 12x + 3 = 9x +8x


23. What is 19% of 3,517? 24. What is 23% of 49?



27. If the population of a town grew 41% up to 7,191. What was the population last year?

Assignment 3.3a

123

28. If the price of an object dropped 35% down to $11.44, what was the original price? Solve.

29. 37 t + 5 = 19 30.

83− (x + 7) = 5 + 3x 31.

32 x – 6 = 7 +

21 x

32. .3(4x + 7) = 2.5x + 6 33. .005x + .045 = .004x 34. 65

47 4 −=+x x

Preparation. 35. Simplify the following (so that there are no negative exponents).

�%��+��,��-��(��.��/��0��1��

Assignment 3.3a

124

Answers:

1. 243m10 28. $17.60

2. x18 29. t = 6

3. f 7 30. x = -

�� 4. t13

31. x = 78

5. �� 32. x = -3

6. 12x8 33. x = -45

7. )�� 34. x = ��

8. 1 35. 5g�h�j k�7a�b�c�d�f < 9. �.��

10. �-�

11. 128m14n35g56

12. 20x9

13. 14

14. 105,146

15. p = 3

16. No solution

17. x = 0

18. x = -1

19. m = <

20. x = ��

21. 115.4%

22. 354.17

23. 668.23

24. 11.27

25. $228

26. $42.74

27. 5,100

Assignment 3.3a

125

Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly, monthly, or daily in some cases. Excel will allow you to make these calculations by adjusting the interest rate and the number of periods to be compounded. Remember that all interest rates provided in the problems are annual rates. You must adjust them to fit other compounding periods. The adjusted rate is called the periodic rate. To adjust the periodic rate in Excel, open the FV calculation box and change a 10% annual rate to quarterly, monthly, or daily as follows:

• Quarterly Rate: .10/4 Changing the rate to 2.5% or .025

• Monthly Rate: .10/12 Changing the rate to .83% or .0083

• Daily Rate: .10/365 Changing the rate to .0274% or .000274

Change ten years of compounding to quarterly, monthly, or daily as follows:

• Quarerly Nper: 10*4 Changing the compounding periods to 40

• Monthly Nper: 10*12 Changing the compounding periods to 120

• Daily Nper: 10*365 Changing the compounding periods to 3,650 If you assume you put $50 into savings and you are comparing savings accounts where the 10% annual interest rate is compounding quarterly, monthly, or daily. You can compare the amount of interest you will earn using Excel as follows: Quarterly Monthly Daily Rate: .1/4 or .025 Rate: .1/12 or .00833 Rate: .1/365or .000274 Nper: 10*4 or 40 Nper: 10*12 or 120 Nper: 10*365 or 3650 Pmt: 0 Pmt: 0 Pmt: 0 Pv: -50 Pv: -50 Pv: -50 Future Value = $134.25 Future Value = $135.35 Future Value = $135.90 The more frequently interest is added to your savings and compounded, the more interest you will earn. The above illustration involves a small amount of savings. The more the savings and the more often you add to your savings the more difference it will make when the interest in added and compounded more frequently. The following example illustrates saving $100 per month for ten years at 10% interest rate compounded monthly versus annually. Annually Monthly Rate: .1 or 10% Rate: .1/12 or .00833 Nper: 10 Nper: 10*12 Pmt: -1200 Pmt: -100 Pv: 0 Pv: 0 Future Value = $19,124.91 Future Value = $20,484.50

126

Savings Plan Formula for a lump sum

= = >�? + AB�BC

Savings Plan Formula with payment

= = >DEF�? + AB�BC − ?AB H

Thus we have the monster formula for a Savings Plan that begins with a balance and then is added to by a payment:

= = >�? + AB�BC + >DEF�? +AB�BC − ?AB H

Spreadsheets normally have this formula built into their functions. It is known as Future Value (FV), so you won’t need to use this one if you learn the spreadsheet well. Loan Payment Formula

>DE = >F AB? − �? + AB�BCH

Spreadsheets also normally have this formula built into their functions. It is known as Payment (PMT). Final note using a spreadsheet: The formulas are built so that money going out from you is negative and money coming in to you is positive. When you are entering Savings into the spreadsheet, the payment and Principal (Present Value) will be negative. However, for a loan, the payment will be negative but the Principal (Present Value) will be positive, because it represents money coming to you.

A = Final Amount PMT = monthly payment P = Principal amount (beginning balance) r = annual interest rate n = number of compounding per year Y = number of years So,

IJ = periodic interest rate (rate

used in spreadsheet)

nY = number of periods (nper)

Section 3.3

127

Calculating Payments, Interest Rates, and Number of Periods Excel will help you calculate the payment you will need to make on a loan. It will calculate the interest rate you would need to earn on your savings to realize a certain future balance. The number of periods it will take to have your savings grow to a certain future balance can also be determined.

Monthly Payment Calculation If you wanted to buy a car that costs $15,000 and you can get a loan at 6% interest for four years, you can determine the monthly payments using the PMT Excel function as follows:

Rate: .06/12 or .005 (monthly interest) Nper: 4*12 or 48 (months) Pv: -15000 Fv: 0 Monthly Payment = $352.28 When you have paid the monthly payment for forty-eight months you will own the car and the future value of the loan is zero because the loan in paid off.

Benefits Versus Bondage You can see how hard your savings will work for you given an interest rate and enough time. However, interest works against you when you borrow money. The benefits may seem great at the moment but the financial bondage is terrible. By calculating the interest you would pay on a loan to borrow a car and the interest you would earn by saving to be able to pay cash for the car, we can determine the financial advantage of collecting interest rather than paying interest. Interest Paid on a Car Loan

You calculate the amount of interest you would pay on a four year car loan of $15,000 at 6% annual interest using the Excel Pmt function as follows:

Rate: .06/12 Nper: 4*12 Pv: -15000 Fv: 0 Monthly Payment = $352.28 Total Payment = $352.28*48 (Payments) = $16,909.22

Section 3.3

128

Interest Paid =$16,909.22 (Paid) -$15,000 (Borrowed) = $1,909.22

TIP: You can have Excel calculate this for you by entering the Pmt function to calculate the monthly payment and then, on the formula bar at the top of the Excel sheet, multiply by 48 payments and subtract the $15,000 you borrowed. The formula will be as follows:

=PMT(0.06/12,4*12,-15000,0)*48-15000

You can also double click on the cell with the Pmt calculation in it and the formula will appear in the cell. Now you can multiply by 48 payments and subtract 15000 and enter this formula in the cell. The cell will have the answer and the formula will be in the formula bar.

Interest Collected on Your Savings

The interest you will earn on your savings of $350.00 per month earning 6% annual interest for 39 months (the number of months we calculated above would be required to accumulate $15,000 in savings) is calculated using the FV function in Excel as follows: Rate: .06/12 Nper: 39 Pmt: -350 Pv: 0

FV = $15,030.44 Amount Deposited in Savings = $350*39 (deposits) = $13,650.00 Interest Earned on Savings = $15,030.44-$13,650.00 = $1,380.44

Again, you can double click on the cell containing the FV calculation and subtract 350*39 and enter this formula giving you the amount of interest earned. You can make the same adjustment to the formula in the formula bar. The resulting formula is as follows:

=FV(0.06/12,39,-350)-350*39 Total Savings From Saving Versus Borrowing

Here is how you benefited by saving and paying cash for the car rather than borrowing the money to buy the car: Interest Earned $1,380.44 Interest Not Paid $1,909.22 Financial Advantage $3,289.66 You are wealthier by $3,289.66 because you collected interest rather than paying interest. This practice will make a major difference in your financial well being throughout your

Section 3.3

129

life. If you put the money you save by paying cash for major purchases to work for you by investing it for your retirement you will add greatly to your independent wealth. You can estimate that using the FV function in Excel as follows assuming a 6% return on your investment for 30 years: Rate: .06 Nper 30 Pv: -3289.66 FV = $18,894.13 This addition to your wealth along with the other additions resulting from saving rather than borrowing will make a major impact on your ultimate wealth.

TIP: In all of the Excel functions you will be using, you only need three entries or factors to calculate the fourth factor you are after. Notice that there are only three entries in each of the above Excel functions. You can leave blank any factor not needed and Excel will assume it is zero.

Section 3.3

130

Section 3.3 Exercises Part B Simplify the following.

1. (3m2)3(2m2)3 2. (x7x11)3

3. �� 4. t8m5t5m3

5. 2-4 6. 3x7 (4x2 – 5x +3)

7. �� 8. � ��

�

��

9. (5p-5g8)-2 10. ��

��

11. ��

��

�� 12. 5x5 (4x7 – 7x6 + 5x-2)

13. Why doesn’t a negative exponent make the answer negative? Using your calculator and the Savings Plan formulas, fill out the table for a savings account.

14. Simple n = 1 15. Quarterly n = 4 16. Monthly n = 12 17. Daily n = 365 P = 200 r = 8% Y = 15 A =

P = 200 r = 8% Y = 15 A =

P = 200 r = 8% Y = 15 A =

P = 200 r = 8% Y = 15 A =

Using your calculator and the Savings Plan formulas, fill out the table for a savings account.


P = 300 r = 7% Y = 15 A =

P = 300 r = 7% Y = 15 A =

P = 300 r = 7% Y = 15 A =

Using a spreadsheet and the Future Value (FV) formula, fill out the table for a savings account. Put your results in a spreadsheet called “Savings and Loan Practice.”


P = 200 r = 7% Y = 15 A =

P = 200 r = 7% Y = 15 A =

P = 200 r = 7% Y = 15 A =

Assignment 3.3b

131

Using a spreadsheet and the Future Value (FV) formula, fill out the table for a savings account. Put your results in a spreadsheet called “Savings and Loan Practice.”


P = 300 r = 8% Y = 15 A =

P = 300 r = 8% Y = 15 A =

P = 300 r = 8% Y = 15 A =

Using your calculator, find the monthly (n = 12) payment for the following loans.

30. P = 300 r = 8% Y = 2 PMT =

31. P = 3000 r = 9% Y = 5 PMT =

32. P = 1500 r = 15% Y = 12 PMT=

33. P = 23,000 r = 8% Y = 30 PMT =

Using a spreadsheet and the Payment (PMT) formula, find the monthly (n = 12) payment for the following loans. Put your results in a spreadsheet called “Savings and Loan Practice.”

34. P = 300 r = 8% Y = 2 PMT =

35. P = 3000 r = 9% Y = 5 PMT =

36. P = 1500 r = 15% Y = 12 PMT=

37. P = 23,000 r = 8% Y = 30 PMT =

Using a spreadsheet and the Payment (PMT) formula, find the monthly (n = 12) payment for the following loans. Put your results in a spreadsheet called “Savings and Loan Practice.”

38. P = 500 r = 4% Y = 2 PMT =

39. P = 4800 r = 9% Y = 5 PMT =

40. P = 2500 r = 15% Y = 12 PMT=

41. P = 23,000 r = 8% Y = 20 PMT =

42. Ensure that every member of the group is able to put in the formulas and use the spreadsheet

to do the calculations.

Assignment 3.3b

132

Answers:

1. 216m12 28. 992.08

2. x54 29. 995.90

3. ��

30. 13.57

4. t13m8 31. 62.28

5. � 32. 22.51

6. 12x9 – 15x8 + 9x7 33. 168.77

7. �� 34. 13.57

8. 1 35. 62.28

9. �!��"

36. 22.51

10. #$��%�! 37. 168.77

11. ��$ "%�� 38. 21.71

12. 20x12 – 35x11 + 25x3 39. 99.64

13. Negative exponents mean division

40. 37.52

14. 634.43 41. 192.38

15. 656.21 42. Complete only when everyone understands and can enter the formulas on their own.

16. 661.38

17. 663.94

18. 827.71

19. 849.54

20. 854.68

21. 857.21

22. 551.81

23. 566.36

24. 569.79

25. 571.47

26. 951.65

27. 984.31

Assignment 3.3b

133

Section 3.3 Exercises Part C – Exam Review Solve.

1. 102

5

135 =

−

−x

2. 2553

2

463 =−

+

+− x

3. -7 – 9m = -22

4. 76 t = -48 5. 19 = 7x - 39 6.

93

74−=

−x


7. V

t

ats=

+

5

2 2

for s 8.

r = pt

I for t

9.

12

2

RR

LRd

+= for L

10. c

gs=

−

11

59 for g

11. 84 is 6 more than 13 times a number. What is the number? 12. Two numbers add to 438 and the first is 72 bigger than the second. What are the two numbers? 14in 13. Find the area of the shaded region: 6in 14. If a rectangle’s length is 5 more than 4 times the width and the perimeter is 180 mm, what are the dimensions of the rectangle? 15. Find the missing variable for a rectangle: P = 78.3 ft w = 17.2 ft l = 16. Find the missing variable for a cylinder: SA = 453.9 in2

r = 7 h = Solve.

17. 7p + 12 = 15 – 7p 18. 3n + 68 = 7 – 2(n – 2) 19. 2x – 10 = 5(x – 4)

Assignment 3.3c

134


22. Original Price:$ 92.56 Tax: 7.3% Final Price:


24. If the price of a meal after a 20% tip was $16.08? What was the price of the meal before the tip was added? 25. If the price of an object dropped 15% down to $413.10, what was the original price? Solve.

26. 25 (-3x+ 2) =

43 x + 6 27.

684

35 +− = xx 28.

73

2172 6 −=+x x

29. Find the price, interest rate and years of a loan for homes in your area. In your “Life Plan” spreadsheet, enter the Price, Number of years, and Interest Rate, then use the PMT formula to figure out how much it will cost to own a home. Report to your group when you have completed it. 30. Using the PMT formula in your “Life Plan” spreadsheet, find the cost of owning your own transportation. Report results to your group. 31. Create a Visual Chart on one side of a piece of paper for Chapter 3 material including information and examples relating to Linear Equations and Applications.

Assignment 3.3c

135

Answers:

1. x = 7 28. x = ��<��

2. x = - �� 29. Will be submitted in Portfolio

3. �� 30. Will be submitted in Portfolio

4. t = -56 31. Make it nice.

5. x = ��

6. x = -5

7. s = �RS%S�

8. T = UVW

9. X = *YZ�[Z�\Z�

10. ] = <^��M�

11. 6

12. 255, 183

13. 55.73in2

14. l = 73mm, w = 17mm

15. 21.95 ft

16. 3.32 in

17. p = ��

18. n = - ��

19. x = �$�

20. 26.5%

21. 425

22. $99.32

23. $21.38

24. $13.40

25. $486.00

26. x = - ��

27. x = -9

Assignment 3.3c

136

Chapter 4

CHARTS, GRAPHS, and LINES

Overview

Algebra 4.1 Charts and Maps 4.2 Lines and Slope 4.3 Writing Equations of Lines

137

Have you ever had difficulty finding locations of objects on maps? If

you haven’t yet had that experience, then I have a little activity for you. At the end of your Bible (King James Version – LDS Edition) there are several maps. On any of the 13 or so maps, try to find the following locations:

Bethsaida Samothrace Iconium Kir-hareseth

Unless you have some help, it might take you a while. Let’s walk through a couple of them together. Right before the maps is an Index of Place-Names. First, we look up Bethsaida. In the edition I have, I find Bethsaida and right next to it is listed 11:C3. The map we have to look at is number 11, but what does the C3 mean? Well, if you turn to map #11 you will notice that across the top are letters and then there are numbers along the side. If you go straight down from C and straight across from 3, you will be right in the vicinity of Bethsaida (right on the north shore of the Sea of Galilee. Next we will look at the Samothrace. In the Index of Place-Names we find that it is located on map 13: E1. Go to map #13. Again, the letters are across the top and numbers are listed on the side. Go straight down from E and across from 1, and you will find a small island with the name of Samothrace. Here is an example from the maps at the end of the Doctrine and Covenants:

Find Harmony, Pennsylvania. In the Index of Place-Names we see that Harmony, Pennsylvania is on two maps, 1:B3 and 3:H3. So we go to map #3 straight down from H and straight across from 3 and find Village of Harmony. If you aren’t familiar with the map, the little grid set up by using one letter and one number is absolutely indispensible. Because letters and numbers go in definite orders, they are called ordinates. When they are used together to pinpoint an exact location, they are then called coordinates. The use of coordinates to find an exactly location was introduced into mathematics centuries ago by a man named Rene Descartes. Now his method to specify locations is used widely in the world. Longitude and latitude are the two numbers that, when used together, can give us an exact location on the planet and form the basis for all ship and plane navigation. Using coordinates is also valuable in being able to read charts and “see” trends that aren’t so readily picked up by only seeing the numbers. Here is an example of a savings account and how it has grown:

Section 4.1 Graphs and Charts

138

Year Amount

2001 $8.31

2002 $17.48

2003 $28.00

2004 $56.39

2005 $72.48

2006 $85.34

The chart helps to visualize the growth. Notice how the graph is made by plotting each set of coordinates. The standard coordinate system that is used in math is called the Cartesian Coordinate

System. It was created by Descartes (hence the name Cartesian) and uses numbers (positive or

negative) for both the horizontal and the vertical measuring. Here is the system and the parts of

it:

Keeping things in order, when you are given a set of numbers like (6,-2), we have the following:

Parentheses tell us that these two numbers go together to make a single set of coordinates

6 comes first and so matches up with the x

-2 comes second and matches up with y

So, we go to where we are at 6 on the x and -2 on the y to find the right location of the point like

this:

$0.00

$20.00

$40.00

$60.00

$80.00

$100.00

2001 2002 2003 2004 2005 2006

Savings Account information

x - axis

tick marks so you won’t lose your place

y - axis

· (6,-2)

Section 4.1

139

Section 4.1 Exercises Part A Find the following locations in the maps section of your Bible using the Index of Place-Names. 1. Marah 2. Haran 3. Mt. Ararat 4. Golgatha Now use the Church History maps. 5. Adam-Ondi-Ahman 6. Nauvoo, Illinois Based on the chart: 7. In what year did the company make $450? 8. How much did the company make in 2006? 9. What years did the company make over $500? 10. Chart the following table of growth of a savings account:

# of

years

0 1 2 3 4 5 6 7 8

Amount 38 50 72 105 130 155 195 205 170

Graph the following points on a Cartesian coordinate system. 11. (1,8) 12. (2,-4) 13. (3,8) 14. (-5,1) 15. (-3,-7) 16. (0,1) 17. (-2,4) 18. (1,-1) 19. (5,0)

0

100

200

300

400

500

600

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Profit

Assignment 4.1a

140

Answers:

1. On maps

2. On maps

3. On maps

4. On maps

5. On maps

6. On maps

7. 2003

8. About $350

9. 2004(maybe), 2009, 2010

10.

11-

19

Assignment 4.1a

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8

Amount

Amount

·(1,8)

·(2,-4)

·(3,8)

·(-5,1)

·(-3,-7)

·(0,1)

·(-2,4)

·(1,-1) ·(5,0)

141

Section 4.1 Exercises Part B Find the following locations in the maps section of your Bible and Doctrine and Covenants (Church History) using the Index of Place-Names. 1. Thessalonica 2. Ur 3. Mt. Nebo 4. Kidron Valley 5. Nineveh 6. Fayette, New York Based on the chart: 7. In what years did the company make about $400? 8. How much did the company make in 2002? 9. What year did the company make less than $375? 10. Chart the following table of growth of a savings account:

# of

years

0 1 2 3 4 5 6 7 8

Amount 238 301 314 325 394 420 447 439 480

11. As a group, use a spreadsheet to make a table of the growth of a savings account for 20 years that begins with $200 and receives a $25 deposit each month and grows at 6%. The table should show the yearly values at the end of each year. 12. As a group, use the spreadsheet to make a graph of the 20-year table. 13. How much money was paid into the savings account over the 20 years? How much interest was earned? Graph the following points on a Cartesian coordinate system. 14. (5,1) 15. (3,-7) 16. (0,-1) 17. (-12,4) 18. (1,1) 19. (-5,0)

0

100

200

300

400

500

600

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Profit

Assignment 4.1b

142

Answers:

1. On maps

2. On maps

3. On maps

4. On maps

5. On maps

6. On maps 11. Answer on next page . . .

7. 2001, 2005, and 2007 12. Answer on next page . . .

8. About $500 13. $6200, 6013.06

9. 2006

10.

14-

19

Assignment 4.1b

·(-5,0)

·(3,-7)

·(0,-1)

·(-12,4)

·(1,1) ·(5,1)

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8

Amount

Amount

143

Answer to #11:

Answer to #12:

t r n P PMT rate nper Amount

1 0.06 12 -200 -25 0.005 12 $520.72

2 0.06 12 -200 -25 0.005 24 $861.23

3 0.06 12 -200 -25 0.005 36 $1,222.74

4 0.06 12 -200 -25 0.005 48 $1,606.54

5 0.06 12 -200 -25 0.005 60 $2,014.02

6 0.06 12 -200 -25 0.005 72 $2,446.63

7 0.06 12 -200 -25 0.005 84 $2,905.92

8 0.06 12 -200 -25 0.005 96 $3,393.54

9 0.06 12 -200 -25 0.005 108 $3,911.24

10 0.06 12 -200 -25 0.005 120 $4,460.86

11 0.06 12 -200 -25 0.005 132 $5,044.39

12 0.06 12 -200 -25 0.005 144 $5,663.90

13 0.06 12 -200 -25 0.005 156 $6,321.63

14 0.06 12 -200 -25 0.005 168 $7,019.92

15 0.06 12 -200 -25 0.005 180 $7,761.29

16 0.06 12 -200 -25 0.005 192 $8,548.37

17 0.06 12 -200 -25 0.005 204 $9,384.01

18 0.06 12 -200 -25 0.005 216 $10,271.18

19 0.06 12 -200 -25 0.005 228 $11,213.08

20 0.06 12 -200 -25 0.005 240 $12,213.06

$0.00

$2,000.00

$4,000.00

$6,000.00

$8,000.00

$10,000.00

$12,000.00

$14,000.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Amount

Amount

Assignment 4.1b

144

Section 4.1 Exercises Part C Find the following locations in the maps section of your Bible and Doctrine and Covenants using the Index of Place-Names. 1. Bethlehem 2. Ephesus 3. Hebron 4. Mt. Sinai 5. Liberty, Missouri 6. Kirtland, Ohio Based on the chart: 7. In what year did the value first drop below 2500? 8. How much value was lost in the 7 years? 9. Which year had the biggest drop in value? 10. Chart the following table of growth of a savings account:

# of

years

0 1 2 3 4 5 6 7 8

Amount 52 35 48 54 62 73 68 72 85

11. Use a spreadsheet to make a table of the growth of a savings account for 20 years that begins with $200 and receives a $50 deposit each month and grows at 6%. The table should show the yearly values at the end of each year. 12.Use the spreadsheet to make a graph of the 20-year table. 13. How much money was paid into the savings account over the 20 years? How much interest was earned? 14. How much interest is earned if the account grows at 9% instead of 6%? Graph the following points on a Cartesian coordinate system. 15. (7,1) 16. (13,-7) 17. (0,4) 18. (8,4) 19. (-1,1) 20. (-8,0)

21. Create a realistic savings plan for yourself. Make a table of the growth of your savings for 20

years. Create a graph of the table. Include it in your portfolio.

Assignment 4.1c

0

500

1000

1500

2000

2500

3000

3500

4000

1 2 3 4 5 6 7

Value

Value

145

Answers:

1. On maps

2. On maps 11. Answer on next page

3. On maps 12. Answer on next page

4. On maps 13. $12200, $11564.09

5. On maps 14. $22,396.17 interest for a

total of $34,596.17

6. On maps

7. 3rd year

8. About $2000

9. From the 1st to 2nd year

10.

15-

20.

21. Include in Portfolio

·(8,4)

·(7,1) ·(-8,1)

·(13,-7)

·(0,4)

·(-1,1)

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Amount

Amount

Assignment 4.1c

146

Answer to number 11:

t r n P PMT rate nper Amount

1 0.06 12 -200 -50 0.005 12 $829.11

2 0.06 12 -200 -50 0.005 24 $1,497.03

3 0.06 12 -200 -50 0.005 36 $2,206.14

4 0.06 12 -200 -50 0.005 48 $2,958.99

5 0.06 12 -200 -50 0.005 60 $3,758.27

6 0.06 12 -200 -50 0.005 72 $4,606.85

7 0.06 12 -200 -50 0.005 84 $5,507.77

8 0.06 12 -200 -50 0.005 96 $6,464.26

9 0.06 12 -200 -50 0.005 108 $7,479.73

10 0.06 12 -200 -50 0.005 120 $8,557.85

11 0.06 12 -200 -50 0.005 132 $9,702.45

12 0.06 12 -200 -50 0.005 144 $10,917.66

13 0.06 12 -200 -50 0.005 156 $12,207.81

14 0.06 12 -200 -50 0.005 168 $13,577.54

15 0.06 12 -200 -50 0.005 180 $15,031.75

16 0.06 12 -200 -50 0.005 192 $16,575.66

17 0.06 12 -200 -50 0.005 204 $18,214.79

18 0.06 12 -200 -50 0.005 216 $19,955.01

19 0.06 12 -200 -50 0.005 228 $21,802.57

20 0.06 12 -200 -50 0.005 240 $23,764.09

Answer to number 12:

Assignment 4.1c

$0.00

$5,000.00

$10,000.00

$15,000.00

$20,000.00

$25,000.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Amount

Amount

147

When we solved equations that looked like 3x-2=13, we got a solution

like x=5, but what does that really mean? We have followed an algorithm to arrive at the proper location, but the reader is reminded that the whole

purpose of manipulating equations is to find numbers for x that we can stick in and make a true statement. If we stick in 5 for x in this equation, we get 3(5) - 2 = 13, which is true. There is no other number which will do this. We call this a solution to the equation.

In that kind of equations we found a number for x that made the statement true, and sometimes we could even guess what would work without really using any formulas or steps. This process becomes a bit helpful when studying the next type of equation:

3x+2y=5

In this type of equation there is an x and a y to find numbers for. The solution to this equation will not be a single number as it was in the earlier cases, but a pair of numbers. The answers will look something like (3,-2), which means that we will stick in 3 for x and -2 for y. If you stick those in, the equation becomes:

3(3)+2(-2) = 5 9-4=5

Woo Hoo! It works! We found a solution, and we don’t even know what we are doing yet. Let’s see if there is another one. Try the following pairs of numbers in the equation to see if they also work:

(1,1) 3(1)+2(1)=5 solution (3,2) 3(3)+2(2) =5 nope (-1,4) 3(-1)+2(4)=5 solution (5,-2) 3(5)+2(-2)=5 nope (5,-5) 3(5)+2(-5)=5 solution If you try all of these, you will realize that some of them work as solutions and some of them don’t. In any case, you should be able to realize that there are a whole lot of solutions, tons of them! One way to get them is to keep guessing. When you get tired of that there is an algorithm that might make things a little easier. If we pick a number for x and stick it in, then we will have an equation that we can solve for y. For example, if we say in this example we want x to be 7, we stick it in to get: 3(7)+2y=5 21+2y=5 2y=-16 y=-8 Which means that when x is 7, y will be -8, or in other words, the pair (7,-8) is a solution. What would we get if we made y = 9? The equation would be: 3x+2(9)=5

Solving for x, we get x= - ��

�, so the pair (-

��

�,9) is a solution.

Now we can get so many solutions this way that it doesn’t pose a problem to find one anymore. Since there are so many, the question arises, “Are there any patterns in the solutions to these equations.” Well, of course there are. This is math! The solutions are pairs, which we can stick on a graph. If we plot the ones that we have already found to the problem we are using we get this: (3,-2), (1,1), (-1,4), (5,-5), (7,-8)

Section 4.2 Graphing

Section 4.2

148

You will notice that all of the solutions are in a straight line. If we connect them, we get all of the solutions for the equation. It is important to realize that if we draw the line that connects the dots, all of the points on that line are solutions. The problems will simply ask you to graph the line 3x+2y=5 or something similar.

The correct answer to

“Graph the line 3x+2y=5”, is then the graph at the left.

For the next one, find four points on the line and then graph it:

y = 41 x – 2

when x = 4 we have y = 41 (4) – 2 which means y = -1.

when x = 0 we have y = 41 (0) – 2 which means y = -2.

when y = 0 we have 0 = 41 (x) – 2 which means x = 8.

when y = 3 we have 3 = 41 (x) – 2 which means x = 20.

The table completely filled out looks like this:

and the graph like this: Notice that we really only need two points to get the pattern. For convenience, we often select 0 for x, and then 0 for y. When x is 0 the point is on the y-axis. Likewise, when y is 0 the point is on the x-axis. In the previous example, the point (0,-2) lies on the y-axis and is called the y-intercept; the point (8,0) lies on the x-axis and is called the x-intercept.

x y

4

0

0

3

x y

4 -1

0 -2

8 0

20 3

An x-intercept happens when y is zero, and a y-intercept happens when x is zero.

.

. .

.

Section 4.2

149

Here is another example. Graph the line 4x + 3y = 8; find the x and y intercepts. We start by finding the x- and y-intercepts with a table that looks like: Then fill it out by plugging in 0 for x and getting 4(0) + 3y = 8 3y = 8

y = 38

When we plug in 0 for y we get: 4x + 3(0) = 8 4x = 8 x = 2 so we have the table:

and the graph: There are a couple of particular kinds of lines that may give you a bit of trouble when you first see them. Your first reaction when asked to graph the line: x = 4 is probably something like, “Hey, where is the y?” or, “How do I do that? It looks different.” Relax, these kind are actually a bit easier than the other ones. Watch: What is x when y is 7? Answer: 4 What is x when y is 0? Answer: 4 What is x when y is -3? Answer: 4 Do you see how nice that is? Since y is not in the equation it can

be anything it wants to be, but x is always 4. The graph is as follows: Here is the line x = 4; notice that it is vertical and hits where x is 4.

x y

0

0

x y

0 38

2 0

The points (4,7) (4,0) and (4,-3) are part of the line and help us graph it.

Section 4.2

150

7%

For future reference you can remember that all equations that only have an x will be vertical. The other special case that may seem difficult at first looks like: y = -2 But I think you can see that it will be very similar to the previous example: What is y when x is 0? Answer: -2 What is y when x is 5? Answer: -2 What is y when x is -3? Answer -2 See how slick that is?! The graph is as follows: Here is the line y = -2; notice that it is horizontal and hits where y is -2. All equations with just a y in them will be horizontal lines.

Now that we can graph any lines, there is one particular property of lines that is most useful. We introduce this by bringing to mind a familiar road sign. This sign warns

of steepness, but take a look at what it is really saying. 7% means the fraction 1007

.The interpretation of the sign means that the road falls vertically 7 feet for every 100 feet that you travel horizontally. In this way the highway department uses fractions to denote the steepness of roads. We are going

to do the same thing with the steepness of lines. When we have a couple of points on the graph we can find the steepness

between them. Here are a couple of examples.

The steepness of the line between point A(-2,1) and B(3,3) is found by

taking how much it changes up and down (distance between 1 and 3 = 2) over how much it changes left and right(distance between -2 and 3 = 5).

That makes a steepness of 52 . The name for steepness is

slope, and the symbol is m (as in a mountain). We would

write that m = 52 .

The slope of the line through A(-2,1) and C(1,7) would be 6 (the distance from 1 to 7)

The points (0,-2) (5,-2) and (-3,-2) are part of the line and help us graph it.

1st Example

2nd Example

A (-2,1)

B (3,3)

C (1,7)

D (5,-3)

Section 4.2

151

over 3 (the distance from -2 to 1.

We would write m = �

�, or in other words m = 2.

The slope of the line through A(-2,1) and D(5,-3) would be -4 (the distance from 1 down

to -3) over 7 (the distance from -2 to 5); m = - 74 .

There are some properties that you should start to see from these examples.

1. Bigger numbers for slope correspond to steeper lines. 2. Positive slopes head up as you go to the right. 3. (Opposite of #2) Negative slopes will head down as you go to the right.

In the first example we obtained the 2 as the distance from 1 to 3. What operation finds distance? Answer: Subtraction. Aha! Seeing that, we can start to see a pattern in how to find slopes a little more quickly. Let’s

look at those three examples, using subtraction this time: Now, see if you can find the slope between two general points: Point 1 and point 2 with coordinates that we don’t know. We would like to call them both just (x,y), but then subtraction would give us zero. This is a good place to introduce you to how subscripts can be very helpful. We will call point #1 (x1,y1) showing that the x and the y come from the 1st point. Similarly we will call point #2 (x2,y2).

Now you can find the slope just like we did in the previous examples:

12

12

xx

yym

−

−=

Voila! You have just created the formula for finding slope between two points. Practice using it quite a bit until it almost becomes natural. Memorize it! Sometimes formulas are written in a few different ways. Here are some of the others:

21

21

xx

yym

−

−=

xchange

ychangem =

run

risem =

x

ym

∆

∆=

They all mean the same thing.

3rd Example

1st Example:

5

2

23

13=

−−

−

3rd Example:

7

4

25

13−=

−−

−−

2nd Example:

23

6

21

17==

−−

−

As a note: You should realize that the subtraction may happen in the opposite direction but will still give the same slope. Example #1 would look like this:

5

2

5

2

32

31=

−

−=

−−

−

Section 4.2

152

Section 4.2 Exercises Part A 1. Two numbers add up to 57, and the first is 23 bigger than the second. What are the two numbers? 2. An international phone call costs 35¢ to connect and 12¢ for every minute of the call. How long can a person talk for $3.60? 3. A 52m rope is cut so that one piece is 18m longer than the other. What are the lengths of the pieces?

4. Original Price:$292.50 Discount:20% Final Price:


6. The perimeter of a rectangle is 82 cm. If the length of the rectangle is 6 more than 4 times the width, what are the dimensions of the rectangle?

Fill out the table for each of the following: Ex. 1 3x + 4y = 7 Solution: 7. x + y = 9 8. 2x – y = 5 9. 5x + 4y = 9 10. x–7y = 13

x y

2

0

1

5

0

x y

5

-4

3

0

7

x y

2

0

-1

0

4

x y

2 41

37 0

1 1

- 413 5

0 47

3(2) + 4y = 7 4y = 1

y = 41

3x + 4(0) = 7 3x = 7

x = 37

3x + 4(1) = 7 3x = 3 x = 1

3(0) + 4y = 7 4y = 7

y = 47

3x + 4(5) = 7 3x = -13

x = - 313

x y

1

0

-3

0

5

x y

1

3

2

0

-1

Assignment 4.2a

153

Graph the following lines, and label three points.

Example: 2x – 7y = 3 Pick three numbers to make a table (intercepts are helpful):

(0,- 73 ) (1,- 7

1 ) (-2,-1)

11. 3x + y = 10 12. y = 2x 13. x – 4y = 7

14. x = 3 15. y = - 73 x + 4 16. 6x – 5y = 12

17. y = -4 18. 5x + 2y = 6 Preparation 19. After reading a bit of section 4.2, try to find the slope between (4,1) and (7,11).

x y

0

1

-2

. . .

Assignment 4.2a

154

Answers:

1. 17, 40 11. (0,10) (3,1) (-1,13) 16. (2,0) (0,-��

) (7,6)

2. 27 minutes

3. 17m, 35m

4. $234

5. $123.17

6. 7cm X 34cm

7.

12. (0,0) (1,2) (2,4) 17. (0,-4) (2,-4) (37,-4)

8.

13. (7,0) (3,-1) (0,��) 18. (0,3) (2,-2) (

� ,0)

9.

14. (3,0) (3,1) (3,2) 19. m = ��

10.

15. (0,4) (7,1) (14,-2)

x y

5 4 -4 13 6 3

9 0

2 7

x y

2 -1 0 -5 -1 -7 �

0

�

4

x y

1 1 0 �

�

-3 6 �

� 0

��

� 5

x y

20 1

34 3

2 ��

�

0 ��

�

6 -1

Assignment 4.2a

155

Section 4.2 Exercises Part B 1. Three types of trees are in a local park. The number of aspens is 4 more than twice as many oaks, and the number of maples is 50 more than the number of oaks. There are a total of 874 trees in the park. How many of each kind are there? 2. If the length is 3 more than 4 times the width of a rectangle and the perimeter is 76mm, what are the dimensions? 3. Solve. 4(x-7) = 2x + 15



6. If my vehicle can get 32 miles per gallon and fuel costs $2.75 per gallon. How many miles per dollar do I get? Fill out the table for each of the following: 7. 2x + y = 9 8. y = 5x+2 9. x + 4y = 9 10. y = �

�x - 13


11. 3x + 2y = 10 12. y = 2x - 7 13. y = ��x

14. x = -6 15. y = - 73 x - 2 16. 2x – 5y = 12

x y

5

-4

3

0

7

x y

2

0

-1

0

4

x y

1

0

-3

0

5

x y

2

5

2

0

-1

Assignment 4.2b

156

17. y = 5 18. 5x + y = 6 Find the slope between each pair of points. Ex. (7,2) (-3,5) 19. (5,-2) (7,3) 20. (4,1) (-5,6) 21. (5,-1) (-3,-8) 22. (7,3) (-2,3) 23. (-5,2) (4,-3) 24. (-6,1) (-6,5) 25. Explain the difference between a slope of zero and an undefined slope. Preparation 26. Find two points of each line and then use those points to find the slope.

2x – 3y = 1 y = ��x + 4

m = 73

25

−−

−

= -10

3

Assignment 4.2b

157

Answers:

1. 205 Oaks, 414 Aspen, 255 Maple

11. (0,5) (��

,0) (2,2) 16. (6,0) (0,-��

) (1,-2)

2. w=7, l=31

3. x = ��

�

4. $314

5. $134.36

6. 11.64 miles per dollar

7.

12. (0,-7) (1,-5) (2,-3) 17. (0,5) (-2,5) (3,5)

8.

13. (0,0) (2, 1) (8,4) 18. (0,6) (1,1) ( � ,0)

9.

14. (-6,0) (-6,1) (-6,2) 19. m = ��

20. m = - ��

21. m = ��

22. m = 0

23. m = -��

24. m = undefined

25. Undefined is straight up and down, vertical. 0 is horizontal, straight across

10.

15. (0,-2) (7,-5) (-7,1) 26. m = �

�

m = �

�

x y

5 -1 -4 17 3 3

� 0

1 7

x y

2 12 0 2 -1 -3

- � 0

� 4

x y

35 2

42 5

2 ��

�

0 -13

28 -1

x y

1 2 0 �

�

-3 3 9 0

-11 5

Assignment 4.2b

158

Section 4.2 Exercises Part C 1. Three types of trees are in a local park. The number of aspens is 4 more than twice the number of birch, and there were 50 more pines than birch. There are a total of 874 trees in the park. How many of each kind are there? 2. If the length is 7 more than 4 times the width of a rectangle and the perimeter is 74mm, what are the dimensions?

3. Solve. 5(x-7) = �

� x + 15



6. What is the Volume of a Cylinder with radius 8cm and height 12cm? Fill out the table for each of the following: 7. 2x + 3y = 9 8. y = -5x+2 9. x - 7y = 9 10. y = �

�x


11. 4x + 2y = 10 12. y = -2x - 7 13. y = ��x

14. x = 5 15. y = - 73 x - 2 16. 7x – 5y = 12

17. y = -3 18. 5x + 2y = 6

x y

5

-4

3

0

7

x y

2

0

-1

0

4

x y

1

0

-3

0

5

x y

2

5

2

0

-1

Assignment 4.2c

159

Find the slope between each pair of points. 19. (4,-2) (7,3) 20. (4,8) (-5,6) 21. (-3,-1) (-3,-8) 22. (7,7) (-2,3) 23. (-5,-3) (4,-3) 24. (-6,1) (-5,5) 25. Explain the difference one more time between a slope of zero and an undefined slope. Find two points of each line and then use those points to find the slope 26. 2x – 3y = 1 27. y = �

�x + 4 28. 5x – y = 10

29. 2x + 7y = 1 30. y = - ��x + 3

Assignment 4.2c

160

Answers:

1. 205 Birch, 414 Aspen, 255 Pine

11. (0,5) (��,0) (1,3) 16. (��

�,0) (0,- ��

�) (1,-1)

2. w = 6, l = 31

3. x = ��

�

4. $74.00

5. $232.40

6. 2412.74 cm3

7.

12. (0,-7) (1,-9) (- �

�,0) 17. (5,-3) (7.2,-3) (0,-3)

8.

13. (0,0) (2, 3) (4,6) 18. (0,3) (2,-2) ( � ,0)

9.

14. (5,2) (5,0) (5,-3.4) 19. m = �

20. m = ��

21. m is undefined

22. m = ��

23. m = 0

24. m = 4

25. Undefined is straight up and down, vertical. 0 is horizontal, straight across

10.

15. (0,-2) (7,-5) (-7,1) 26. (0,- �) (�

� , 0) m = �

27. (0,4) (5,7) m = �

28. (2,0) (0, -10) m = 5

29. (0, ��) (�

�,0) m = - �

�

30. (0,3) (7,1) m = - ��

x y

5 -

�

-4 �

�

0 3

�

� 0

-6 7

x y

2 -8 0 2 -1 7

�

� 0

- �

� 4

x y

1 - �

�

0 - �

�

-3 - ��

�

9 0

44 5

x y

��

� 2

��

� 5

2 �

�

0 0

-�

� -1

Assignment 4.2c

161

(2,1)

(6,4)

Okay, now that you know how to graph a line by getting some points, and you know how to find the slope between two points, you should be able to find the slope of a line once you have an equation:

Example: Find the slope and graph the line 3x-4y=2 Well, if we find a couple of points: (2,1) and (6,4), the graph must look like this: Then finding the slope, we can just use the same method that we have done the other ones we get the slope

43

2614 ==

−

−m .

Trying this a couple of times on various equations, you might get something like the following:

I hope that you kind of see a pattern emerging that you would be able to use as a shortcut. Do you see how the change in y is always the coefficient of x? And do you see that the change in x is always the opposite of the coefficient of y? These equations are all written the same way and

have the same pattern for getting the slope without actually figuring it out from a couple of points. Here is a pattern for another common way of writing lines. Pick out the pattern here:

The pattern here is even easier than the first one. When y is by itself, the slope is simply the number in front of x. No change.

Equation Slope:

3x – 5y = 10 m= 53

2x + 9y = 4 m = - 92

5x + y = 15 m = -5

x-3y = 12 m = 31

Equation: Slope:

y = -2x – 5 m = -2

y = 73 x + 4 m = 7

3

y = - 94 x – 13 m = - 9

4

y = - 2 m = 71 7

x

Section 4.3 Graphing Equations with Slope

Section 4.3

162

3

-5

Since these are two very common ways of writing lines, they need some comparison.

Standard Form: - It is written in the form Ax+By = C where

A, B, and C are integers (usually). - Intercepts are found by putting in 0 for

either x or y so each is relatively easily

found as (0,B

C ) and (A

C ,0).

- Slope is always m = -B

A

Advantages: It has no fractions. Both x- and y- intercept have same amount of calculation Disadvantages: Minor calculation for y-intercept. Remembering to put the negative sign on the slope.

Slope-intercept Form: - Written in the form y = mx+b. - m is the slope without any adjustment. - (0,b) is the y-intercept.

Advantages: Slope is most easily found. Y-intercept is most easily found. Prepares you for function f(x) notation. Disadvantages: Fractions are often part of the equation.

Since both of them will be given to you to graph, you should be able to work with both of them. Important Note: You should also see that we can change from Standard form into Slope-intercept form (and vice-versa) simply by solving for y. In the example 3x-4y=2, we get: 3x-4y=2 -4y=-3x+2

y= 43 x - 2

1

They are simply two different ways to write the same equation. There is no difference, except that of convenience. The first way is called standard form, and the second is called slope-intercept form. Again, they are the same line! Every point that works in one will work in the other. In any case, you will learn and have practice with both forms. Being able to pick out intercepts and slope from lines will help you to graph them quickly. Having the slope especially makes it a cinch to graph lines. You only need to find one point, then follow the slope to the next point and draw the line. Example:

Graph the line and find the slope of y=- 35 x - 4

Standard: 3x-4y=2 Slope-intercept: y= 43 x - 2

1

Slope: m= 43 Slope: m= 4

3

y-intercept: (0, - 21 )

x-intercept: ( 32 ,0)

y-intercept: (0, - 21 )

x-intercept: ( 32 ,0)

Section 4.3

163

Well the slope is right in front of x, so m= - 35

One easy point is to stick in zero for x. We get the point (0,-4). Following the slope, (it is negative, so we will head down as we go to the right) down 5 over 3 and we come to the point (3,-9), and then draw the line.

Another example: Graph the line and find the slope of 2x-7y=4

Well the slope is the opposite of 2 over -7, so m=- 72

−= 7

2

It appears that the easiest point in this one is the x-intercept, so we stick in zero for y and get x=2: (2,0). Following the slope we move up 2 and over 7 to the next point (9,2), and then draw the line. That covers graphing and finding the slope for the vast majority of equations. As you will recall, there were a couple of special cases

where either the x or the y were missing. We now look to find the slope of these. We will work two examples of this: First: y= -2 Remember how to find a couple of points that work: (3,-2) and (-1,-2). It gives us the graph of a horizontal line where y is always -2: Putting those two points in to the formula for finding slope, we get:

m = 04

0

31

22=

−=

−−

−−−

which means that all horizontal lines will have a slope of 0.

Second: x=5 Remember how to find a couple of points that work: (5,2) and (5,6). It gives us the graph of a vertical line where x is always 5: Now if we put the points in the slope formula, we get:

55

26

−

− =

0

4= bad news. (Division by zero is

undefined.) which means that all vertical lines have undefined slope.

2

7

Section 4.3

164

2=m

1=m

21=m

31=m

0=m

15=m

2−=m

1−=m

32−=m

51−=m

8−=m

m = undefined

m = undefined

To get a feel for slope a little bit better, we are going to take a little time to look at some slopes. You will notice that

the higher the number, the steeper it is. Common sense from that will tell you that a slope of 0 will belong to a line that is completely flat. Also, you

should see that since numbers get bigger as the slope gets steeper, the slope of a vertical line would have to be far greater than a billion.

On the other hand, numbers get increasingly large in the negative direction for lines that are heading down ever steeper. That means that vertical lines would have to have a slope that is less than negative one billion. Hmmmmmm…. greater than a billion and

less than negative a billion at the same time. No wonder that division by zero can’t be done and is undefined.

A word of caution: Since the term “no slope” is interpreted by some to mean zero slope and by

others to mean that the slope doesn’t exist, we will simply avoid the term. A vertical line has undefined

slope and a horizontal line has a slope of zero.

Section 4.3

165

2 -2

7

7

a

b

Now that we can go from the equation of a line to the finding of points, getting the slope and

graphing the line, we are going to work on how to go backwards. It really isn’t as difficult as it

seems. Since from an equation we can get the slope, we can certainly write an equation from the slope. Example:

Write an equation of the line that has slope m= 53 , and goes through the point (5,2).

There are two ways to do this. Remember with standard form, the slope is the negative of the first number over the second. In slope-intercept, the slope is right in front of the x when y is by itself.

Standard Slope-Intercept

Since the slope is 53 , we know that an

equation would have to be: 3x – 5y = something but what? Ahhh, here is where we use the fact that (5,2) has to work in the equation. If we stick in 5 for x and 2 for y we get: 3(5) – 5(2) = 15 – 10 =5 3x – 5y = 5

Since the slope is 53 , we know that an

equation would have to be:

y = 53 x+b but what is b? Ahhh, here

is where we use the fact that (5,2) has to work in the equation. If we stick in 5 for x and 2 for y we get:

2 = 53 (5) + b

2 = 3 + b -1 = b Thus our equation must be

y = 53 x - 1

As a side note on slope: When two lines have the same slope, or steepness such that they never cross, we call these parallel. When two lines meet at a 90 degree angle, it is called perpendicular. Let’s suppose that line “a”

has a slope of 27 ; it is pretty steep and positive. Line

“b” will have a similar ratio, but we can see that it is shallow and negative. As we can see from the

picture, the slope of “b” is m= - 72 . This pattern

happens every time that two lines are perpendicular. As a rule: A perpendicular slope is the negative reciprocal. Some people like to think of it that the

Again, please note that if you take the standard form and solve for y, you will get the slope-intercept

form.

Section 4.3

166

two slopes will always multiply together to give you -1. As a special case, can you see what slope would be perpendicular to 0? Since vertical and horizontal are perpendicular, an undefined slope is the answer. Here are a few examples:

Equation Slope Parallel slope Perpendicular

3x+2y=7 m= - 23 m= - 2

3 m= 32

y = 5x-2 m=5 m=5 m= - 51

4x-7y=7 m= 74 m= 7

4 m= - 47

x=-7 m = undefined m = undefined m=0

y=3 m=0 m=0 m = undefined

167

Section 4.3 Exercises Part A 1. Three types of horses are in a local ranch. The number of Arabians is 8 more than twice the number of Quarter-horses, and the number of Clydesdales is50 more than the number of Quarter-horses. There are a total of 282 horses at the ranch. How many of each kind are there? 2. What is the slant height of a cone that has Surface Area of 219.91 in2 and a radius of 5 in? 3. The perimeter of a rectangle is 120 in. If the length of the rectangle is 3 more than twice the width, what are the dimensions of the rectangle?

4. Original Price:$392.50 Tax: 6% Final Price:


Fill out the table for each of the following: 6. 2x - 3y = 9 7. y = �

�x+2

Graph the following lines, and label x and y intercepts.

8. 5x + 2y = 10 9. y = ��x - 6 10. y = �

�x

11. x = 10 12. y = - 73 x +4 13. 7x – y = 14

Find the slope between each pair of points. 14. (8,-2) (7,3) 15. (8,1) (-5,6) 16. (-3,-1) (-3,-8) 17. (7,9) (-2,3) 18. (-5,2) (4,6) 19. (-6,1) (6,1)

x y

5

-4

3

0

7

x y

2

0

-1

0

4

Assignment 4.3a

168

Graph the following lines giving one point and the slope. Ex. 2x – 7y = 3

Find one point: ( 23 ,0) and the slope: m = 7

2 .

Then graph the point. Then go up 2 and over 7 for the next one: 20. -6x + y = 10 21. y = 4x + 3 22. y = �

�x - 4

23. x = -6 24. y = - 73 x - 2 25. 3x – 4y = 12

26. 5x + 3y = 10 27. x + 4y = 9 28. y = 7 Preparation 29. Write down 5 equations of lines that have the slope:

m= - �

�

. 2 7

Assignment 4.3a

169

Answers:

1. 56 Quarter-horses, 106 Clydesdales, 120 Arabian

10. (0,0) (3,8) 20. (0,10); m = 6

2. slant height = 9 in

3. 41in X 19in

4. $416.05

5. $69.07 11. (10,0) no y-int 21. (0,3); m = 4

6.

7.

12. (0,4) ( ��

,0) 22. (0,-4); m = ��

8. (0,5) (2,0)

13. (2,0) (0,-14) 23. (-6,0); m = undefined

9. (0,-6) ( ��

,0)

14. m = -5 24. (0,-2); m = - ��

15. m = - ��

16. m = undefined

17. m = ��

18. m = ��

19. m = 0

x y

5 ��

-4 -��

9 3

� 0

15 7

x y

2 9 0 2 -1 - �

- �� 0

�� 4

Assignment 4.3a

170

25. (4,0); m = ��

27. (9,0); m = - �� 29. Discuss it together.

26.

(2,0); m = - �� 28. (15,7); m = 0

Assignment 4.3a

171

Slope Monster Equation Slope Equation Slope

2x – 5y = 7 4x – y = 7

y = �

�x - 4 y =

�

�x - 4

5x – 3y = 7 8x – 3y = 12

2x + 7y = 19 - 4x + 7y = 19

x = 13 x = -19

y = �

�x - 8 y =

�

�x - 4

y = 5x – 8 y = -3x – 8

-3x + 9y = 4 -10x + 6y = 4

y = -3 y = 15

y = - �

��x - 4 y =

�

��x - 4

7x – 3y = 7 2x – 8y = 17

y = �

�x - 4 y =

�

�x + 6

5x – 3y = 7 4x + 7y = 7

4x + 7y = 19 2x - 9y = 19

x = - 3 x = 7

y = - �

�x - 4 y =

�

�x - 4

y = -2x – 8 y = 4x + 13

-3x + 6y = 4 -3x - 6y = 4

y = -5 y = 7

y = - �

�x - 4 y = -

�

�x + 15

Assignment 4.3 Slope Monster

172

Slope Monster Solution Equation Slope Equation Slope

2x – 5y = 7 m = �

� 4x – y = 7 m = 4

y = �

�x - 4 m =

�

� y =

�

�x - 4 m =

�

�

5x – 3y = 7 m = �

� 8x – 3y = 12 m =

�

�

2x + 7y = 19 m = −�

� - 4x + 7y = 19 m =

�

�

x = 13 Undefined x = -19 Undefined

y = �

�x - 8 m =

�

� y =

�

�x - 4 m =

�

�

y = 5x – 8 m = 5 y = -3x – 8 m = -3

-3x + 9y = 4 m =

� -10x + 6y = 4 m =

�

�

y = -3 m = 0 y = 15 m = 0

y = - �

x - 4 m = −

�

y =

x - 4 m =

7x – 3y = 7 m = �

� 2x – 8y = 17 m =

�

y = �

�x - 4 m =

�

� y =

�

�x + 6 m =

�

�

5x – 3y = 7 m = �

� 4x + 7y = 7 m = −

�

�

4x + 7y = 19 m = −�

� 2x - 9y = 19 m =

�

�

x = - 3 Undefined x = 7 Undefined

y = - �

�x – 4 m = −

�

� y =

�

�x - 4 m =

�

�

y = -2x – 8 m = - 2 y = 4x + 13 m =4

-3x + 6y = 4 m =

� -3x - 6y = 4 m = −

�

y = -5 m = 0 y = 7 m =0

y = - �

�x - 4 m = −

�

� y = -

�

�x + 15 m = −

�

�

173

Section 4.3 Exercises Part B Fill out the table for each of the following: 1. 2x - 5y = 11 2. y = �

�x+6


3. 4x - 2y = 10 4. y = - ��x - 6 5. y = 5x

Find the slope between each pair of points. 6. (3,-2) (7,3) 7. (9,1) (-7,6) 8. (5,-1) (-3,-8) 9. (-2,9) (-2,3) 10. (-5,2) (5,6) 11. (19,1) (6,1) 12. Explain the difference between a slope of zero and an undefined slope. Graph the following lines giving one point and the slope. 13. -3x + 4y = 10 14. y = 2x - 7 15. y = �

�x - 4

16. y = 17 17. y = - 73 x - 2 18. 2x – 6y = 12

Write the equations of the lines with the slopes and points: Ex.

Write an equation of the line that has slope m = 74 , and goes through the point (2,1). Put the

answer in Standard Form.

From the slope m = 74 , I know that the equation must look like:

4x – 7y = something, so I put in the point to see what it is. 4(2) – 7(1) = 1.

x y

5

-4

3

0

7

x y

2

0

-1

0

4

Assignment 4.3b

174

Thus the answer is 4x – 7y = 1. Ex.

Write an equation of the line that has slope m = 74 , and goes through the point (2,1). Put the

answer in Slope-Intercept Form.

From the slope m = 74 , I know that the equation must look like:

y = 74 x + b Put the point in to see what b is.

1 = 74 (2) + b

1- 78 = b

- 71 = b

Thus the answer is y = 74 x - 7

1 .

19. Write an equation of the line that has slope m= -3, and goes through the point (-4,6). Put the answer in Standard Form.

20. Write an equation of the line that has slope m= 85 , and goes through the point (3,6). Put the


21. Write an equation of the line that has slope m=- 32 , and goes through the point (1,-3). Put the


22. Write an equation of the line that has slope m=- 54 , and goes through the point (5,-3). Put the

answer in Slope-Intercept Form. 23. Write an equation of the line that has slope m= 2, and goes through the point (0,5). Put the answer in Slope-Intercept Form.

24. Write an equation of the line that has slope m=- 71 , and goes through the point (-4,7). Put the


Assignment 4.3b

175

Answers: 1. 6. m = �

� 18. (6,0) m = �

�

7. m = - ��

8. m = ��

9. m = undefined 10. m = �

�

11. m = 0 12. Undefined is vertical

0 is horizontal

13. (0,��) m = �

� 19. 3x + y = -6

2. 20. 5x – 8y = -33 21. y = - �

�x - �

�

22. y = - ��x + 1

23. y = 2x + 5 24. x + 7y = 45 14. (0,-7) m = 2 3. (0,-5) (�

�,0)

15. (0,-4) m = �

�

4. (0,-6) (-��

�,0)

16. (0,17) m = 0 5. (0,0) (2,10) 17. (0,-2) m = -�

�

x y

5 - �

-4 - ��

13 3

0

23 7

x y

2 13

0 6

-1 ��

- ��

0

- �� 4

Assignment 4.3b

176

Section 4.3 Exercises Part C – Exam Review 1. Chart the following table of growth of a savings account:

# of years

0 1 2 3 4 5 6 7 8

Amount 35 50 75 102 130 161 174 205 240

Based on the chart: 2. In what year did the company make $450? 3. How much did the company make in 2006? 4. What years did the company make over $500? 5. Graph the growth of a savings account in Excel, using the savings formula, over the course of 20 years of a savings account that starts out at $200 and adds $50 per month and gets 7% interest. Fill out the table for each of the following: 6. 3x + 4y = 11 7. y = �

�x - 2


8. 5x - y = 10 9. y = - ��x - 5 10. y = -2x

0

100

200

300

400

500

600

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Profit

x y

5

-4

3

0

7

x y

2

0

-1

0

4

Assignment 4.3c

177

Find the slope between each pair of points. 11. (4,-2) (7,3) 12. (3,1) (-7,6) 13. (5,-1) (5,-8) 14. (-2,9) (-2,254) 15. (-5,2) (5,7) 16. (19,1) (6,2) 17. Explain the difference (again) between a slope of zero and an undefined slope. Graph the following lines giving one point and the slope. 18. -3x + 5y = 10 19. y = - x - 2 20. y = �

�x - 1

21. y = -5 22. y = - �

� x - 2 23. 12x – 6y = 12

Write the equations of the lines with the slopes and points: 24. Write an equation of the line that has slope m = 2, and goes through the point (-4,1). Put the answer in Standard Form.

25. Write an equation of the line that has slope m = �

� ,and goes through the point (-14,6). Put the


26. Write an equation of the line that has slope m = - 32 , and goes through the point (0,0). Put the


27. Write an equation of the line that has slope m = - �

� , and goes through the point (2,-3). Put

the answer in Slope-Intercept Form. 28. Write an equation of the line that has slope m = 7, and goes through the point (0,5). Put the answer in Slope-Intercept Form.

29. Write an equation of the line that has slope m = - �

�, and goes through the point (-4,7). Put

the answer in Standard Form. 30. Create a Visual Chart on one side of a piece of paper for Chapter 4 material including information and examples relating to Charts, Graphs, and Lines.

Assignment 4.3c

178

Answers: 1. 20. (0,-1) m = �

�

9. (0,-5) (-��

,0)

21. (0,-5) m = 0 10. (0,0) (2,-4) 2. 2003 3. about $340 4. 2004(?), 2009, 2010 5. On Spreadsheet 22. (0,-2) m = -

�

6. 11. m = �

�

12. m = - ��

13. m = undefined 14. m = undefined 23. (1,0) m = 2 7. 15. m = �

�

16. m = - �

��

17. Undefined is vertical 0 is horizontal

18. (0,2) m = ��

8. (0,-10) (2,0) 24. 2x – y = -9 25. 3x – 7y = -84 19. (0,-2) m = -1 26. y = - �

�x

27. y = - ��x + 2

28. y = 7x + 5 29. x + 4y = 24 30. Make it nice.

0

100

200

300

0 1 2 3 4 5 6 7 8

x y

5 -1 -4 �

�

- �

� 3

��

� 0

-��

� 7

x y

2 �

0 -2 -1 -��

�

�� 0

��

4

Assignment 4.3c

179

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Libro de Algebra