LESSON Rational Numbers 2-1lfandres.weebly.com/uploads/2/5/8/0/25801520/pre_alg_2_chapter_notes.pdf" $ 10 3"1 1 0 " # " 3(10 1) 0 $ 1 " 10 10 # 10 Add numerators. Keep the denominator

Rational Numbers2-1

LESSON

Lesson ObjectivesWrite rational numbers in equivalent forms

Vocabulary

rational number (p. 68)

relatively prime (p. 68)

Additional Examples

Example 1

Simplify.

A. !1860! 16 " 1 ! 16 is a common factor.

80 " 5 ! 16

!1860! " 16 #

80 #Divide the numerator and denominator by .

"

B. !$2198

! 18 " 2 ! 9 There are no factors.

29 " 1 ! 29

!$2198

! " $18 and 29 are prime.

Example 2

Write each decimal as a fraction in simplest form.

A. 5.37

" is in the hundredths place.

Copyright © by Holt McDougal. 28 Holt McDougal MathematicsAll rights reserved.


LESSON 2-1 CONTINUED

B. 0.622

! 1000 2 is in the place.

! Simplify by by the common factor 2.

Example 3

Write each fraction as a decimal.

A. "191"

The pattern repeats, so draw a bar over the 2 to indicate that

this is a decimal.

The fraction "191" is equivalent to the decimal .

B. "270"

This is a decimal.

The remainder is .

The fraction "270" is equivalent to the decimal .

Check It Out!

1. Simplify.

"1287"

2. Write the decimal as a fraction in simplest form.

8.75

3. Write the fraction as a decimal.

"195"

Copyright © by Holt McDougal. 30 Holt McDougal MathematicsAll rights reserved.Copyright © by Holt McDougal. 30 Holt McDougal MathematicsAll rights reserved.

Comparing and Ordering Rational Numbers2-2

LESSON

Lesson ObjectivesCompare and order positive and negative rational numbers written as fractions,decimals, and integers

Vocabulary

least common denominator (LCD) (p. 72)

Additional Examples

Example 1

Compare. Write !, ", or #.

A. !56! !1

70!

Method 1 Multiply to find the denominator.

6 ! 10 " Multiply and to find a common denominator.

!56! ! " !

56

!!

1100! " Write the fractions with a common

denominator.

!170! ! " !1

70!!66! "

!5600! !

4620!, so !

56! !1

70!



B. !23! !

45!

Method 2 Find the least common denominator.

3: 3, 6, 9, 12, 15, . . . 5: 5, 10, 15, . . . List multiples of 3 and 5. The LCM

is .

!23! ! " !

23

!!

55! " Write the fractions with a common

denominator.

!45! ! " !

45

!!

33! "

!1105! !

1125!, so !

23! !

45!

Example 2


A. 5!29! 5!

27!

5!29! " and 5!

27! " Write the fractions as .

5.2! 5.286, so 5!29! 5!

27!

B. #0.44 #!25!

#!25! " Write as a .

#0.44 #0.4, so #0.44 #$25$

C. !19! 0.1

!19! " Write as a .

0.1! 0.1, so !19! 0.1



Example 3

The numbers !144!, !3.4, 6.0, and !2.5 represent the percentage of change

in populations for four states. List these numbers in order from least togreatest.

Place the numbers on a number line and read them from left to right

The percent changes in population from least to greatest are "3.4, "2.5, !144!, and 6.0.

Check It Out!

1. Compare. Write ", #, or $.

!13! !

12!

2. Compare. Write ", #, or $.

"0.64 "!160!

3. The numbers !1.2, !73!, and 2.3 represent the percent of change in

population for three states. List these numbers in order from least togreatest.

5432 610!5 !4 !3 !2 !1


Adding and Subtracting Rational Numbers2-3

LESSON

Lesson ObjectivesAdd and subtract decimals and rational numbers with like denominators

Additional Examples

Example 1

In August 2001 at the World University Games in Beijing, China, JimyriaHicks ran the 200-meter dash in 24.08 seconds. Her best time at the U.S.Senior National Meet in June of the same year was 23.35 seconds. Howmuch faster did she run in June?

24.08!23.35 Align the decimals.

She ran second faster in June.

Example 2

Use a number line to find each sum.

A. 0.3 " (!1.2)

Move right units.

From , move left

units.

You finish at , so 0.3 " (!1.2) # .

B. !15! " !25!

Move right units.

From , move right

units.

You finish at , so $15$ " $

25$ # .

0 1 125

15

15

35

45

0.40!0.4!1.0!1.4



Example 3

Add or subtract. Write each answer in simplest form.

A. !"29" ! "

59"

!"29" ! "

59" # """"""""9 # Subtract numerators. Keep

the denominator.

B. "67" $ !!"

37""

"67" $ !!"

37"" # 7 # Add numerators. Keep

the denominator.

Example 4

Evaluate each expression for the given value of the variable.

A. 12.1 ! x for x # !0.1

12.1 ! ( ) Substitute for x.

Think: 12.1 ! (!0.1) # 12.1 0.1

B. "170" $ m for m # 3"1

10"

"170" $ Substitute for m.

"170" $ 10 3"1

10" # "

3(101)0$ 1" 10

10 # 10 Add numerators. Keep the denominator.

Check It Out!

1. Tom ran the 100-meter dash in 11.5 seconds last year. This year heimproved his time by 0.568 seconds. How fast did Tom run the 100-meter dash this year?


Multiplying Rational Numbers2-4

LESSON

Lesson ObjectivesMultiply fractions, mixed numbers, and decimals

Additional Examples

Example 1

Multiply. Write each answer in simplest form.

A. !8!"67""

# "!8

7! 6"

# Multiply.

# Simplify.

B. 2!5"13""

2! 3 " 5"13" # 3 # 3

3 Multiply.

3 Multiply.

Simplify.



Example 2


A. !18!!!

67!"

" !18((67))! Multiply .

Multiply .

" !18((67))! Look for common : 2.

" Simplest form

B. #!23!!!

92!"

" !#32(2(9))

! Multiply .

Multiply .

" !#32(2(9))

! Look for common : 2, 3.

" 1 " Simplest form

Example 3

Multiply.

A. 2(#0.51)

2 ! (#0.51) " Product is with

2 decimal places.

B. (#0.4)(#3.75)

(#0.4) ! (#3.75) " Product is with

decimal places.

" You can drop the zeros after the 5.



Example 4

Joy completes !210! of her painting each day. How much of her painting

does she complete in a 7-day week?

!210!(7) " !2

10!! " Write as an improper fraction.

" 20 ! 1 Multiply

"

Joy completes of her painting in a 7-day week.

Check It Out!

1. Multiply. Write the answer in simplest form.

#3!!58!"

2. Multiply. Write the answer in simplest form.

!35!!!

58!"

3. Multiply.

3.1(0.28)

4. Ken completes !110! of his work each day. How much work does he

complete in a 5-day week?


Dividing Rational Numbers2-5

LESSON

Lesson ObjectivesDivide fractions and decimals

Vocabulary

reciprocal (p. 84)

Additional Examples

Example 1

Divide. Write each answer in simplest form.

A. !151! " !

12!

!151! " !

12! # !1

51! ! Multiply by the .

# !151! ! No common

# Simplest form

B. 2!38! " 2

2!38! " 2 # " !

21! Write as an fraction.

# !189! Multiply by the .

# !189!!21

! No common .

# # 19 " 16 # 1 R

Example 2

Find. 0.384 ! 0.24

0.384 " 0.24 # !00.3.2844

! !!110000!" #

# Divide.



Example 3


A. !!5.

n25! for n " 0.15

!50..2155! " !

50..2155! !!110

000!" 0.15 has 2 decimal places, so use .

" Divide.

"

B. k # !45! for k " 5

5 # !45! " !

51! !

" !!1 ! 45 ! 5 " "

Example 4 PROBLEM SOLVING APPLICATION

A cookie recipe calls for !12! cup of oats. You have !34! cup of oats. Howmany batches of cookies can you bake using all of the oats you have?

1. Understand the ProblemThe number of batches of cookies you can bake is the number of batchesusing the oats that you have. List the important information:

• The amount of oats is cup.

• One batch of cookies calls for cup of oats.

2. Make a PlanSet up an equation.

amount of oatsyou have

amount forone batch

number ofbatches" #



3. SolveLet n ! number of batches.

"34" # "

12" ! n

"34" ! ! n

"64", or batches of the cookies.

4. Look BackOne cup of oats would make batches so 1"

12" is a

answer.

Check It Out!

1. Divide. Write the answer in simplest form.

4"25" # 3

2. Find. 0.585 ! 0.25.

3. Evaluate the expression for the given value of the variable.

"2.

b55" for b ! 0.75

4. A ship will use "16" of its total fuel load for a typical round trip. If there is "58" of a total fuel load on board now, how many complete trips can be made?


Adding and Subtracting with UnlikeDenominators2-6

LESSON

Lesson ObjectivesAdd and subtract fractions with unlike denominators

Additional Examples

Example 1

Add or Subtract.

Method 1:

A. !18! " !

27! Find a common :

8(7) # .

# !18! ! " " !

27! ! " Multiply by fractions equal to .

# " Rewrite with a common

.

# Simplify.

Method 2:

B. 1!16! $ 1!

58!

# $ !183! Write as an fraction.

Multiples of 6: 6; 12; 24; 30 List the of each

Multiples of 8: 8; 16; 24; 32 denominator and find the .

# !76! $ !

183! Multiply by fractions equal to .

# !!!!24$ !!!!24 # $!!!!24

Rewrite with a denominator.

# Simplify.



Example 2

Evaluate t ! !45! for t " !

56!.

t " !45!

# " !45! Substitute for t.

# !56! " !

45! Multiply by fractions equal to .

# " Rewrite with a denominator:

6(5) # .

# Simplify.

Example 3

Two dancers are making necklaces from ribbon for their costumes. They need pieces measuring 13!

34! inches and 12!

78! inches. How much ribbon

will be left over after the pieces are cut from a 36-inch length?

13!34! $ 12!

78! Add to find the length of the two pieces.

# 4 $ 8 Write as improper fractions. The LCD is 8.

# 8 $ !1083

! Rewrite with common denominators.

# 8 , or

The length of the two ribbons is 26!58! inches. Now find the amount of ribbon

left over.



36 ! 26"58" Subtract the length of the two ribbons from

the total length of ribbon.

# 36 ! 8 Write as improper fractions. The LCD is 8.

# 8 ! "2183

" Rewrite with common denominators.

# 8 , or

There will be inches left.

Check It Out!

1. Add.

2"16" $ "

34"

2. Evaluate !59! ! h for h " !!127!.

3. Michael wants to run 10 miles in three days. On Monday, he ran 2"

34" miles. On Tuesday, he ran 3"

13" miles. How many miles does Michael

need to run on Wednesday?


Solving Equations with Rational Numbers2-7

LESSON

Lesson ObjectivesSolve equations with rational numbers

Additional Examples

Example 1

Solve.

A. m ! 4.6 " 9

m ! 4.6 " 9

# " # Subtract from both sides.

m "

B. 8.2p " #32.8

$$$8.2p

" $$$#32.8

Divide both sides by .

p "

C. $1x.2$ " 15

! $1x.2$ " ! 15 Multiply both sides by .

x "

Example 2

Solve.

A. n ! $27$ " #$

37$

n # ! $27$ " #$

37$ # Subtract from both sides.

n "



3

4

B. y ! "16" # "

23"

$ y ! "16" # "

23" $ Add to both sides.

y # 6 $ 6 Find a common ; 6.

y # Simplify.

C. "56"x # "

58"

"65" ! "

56"x # "

58" ! "

65" Multiply both sides by .

x # Simplify.

Example 3

Mr. Rios wants to prepare a dessert, but only has 2"23" tablespoons of

sugar. If each serving of the dessert has "23" tablespoon of sugar, how

many servings can he make for the party?

Convert fractions: 2"23" # "

3(2)3$ 2" #

Write an equation:

! #

"23" ! x # "

83"

Now solve the equation.

"23" ! x # "

83"

"23" ! ! x # "

83" ! Divide both sides by . Multiply

by the reciprocal, .

x # , or Simplify.

Mr. Rios can make servings.

Total amount ofsugar

Total number ofservings

Amount of sugar in one serving



Check It Out!

1. Solve.

m ! 9.1 " 3

2. #38#x " #196#

3. Rick’s car holds 23! the amount of gasoline as his wife’s van. If the car’sgas tank can hold !31

2! gallons of gasoline, how much gasoline can thetank in the minivan hold?


Solving Two-Step Equations2-8

LESSON

Lesson ObjectivesSolve two-step equations

Additional Examples

Example 1 PROBLEM SOLVING APPLICATION

The mechanic’s bill to repair Mr. Wong’s car was $650. The mechaniccharges $45 an hour for labor, and the parts that were used cost $443.How many hours did the mechanic work on the car?

1. Understand the ProblemList the important information:

The answer is the number of the mechanic worked on the car.

• The parts cost $ .

• The labor cost $ per hour.

• The total bill was $ .

Let h represent the hours the mechanic worked.

Total bill ! Parts " Labor

! " h

2. Make a PlanThink: First the variable is multiplied by , and then is added

to the result. Work backward to solve the equation. Undo the operations in

reverse order: First subtract from both sides of the equation, and

then divide both sides of the new equation by .



3. Solve

! " h

# # Subtract to undo the addition.

! h

! Divide to undo multiplication.

! h

The mechanic worked for hours on Mr. Wong’s car.

4. Look Back

If the mechanic worked hours, the labor would be $45(4.6) ! $207.

The sum of the parts and the labor would be $ " $ !

$ .

Example 2

Solve.

A. $n3$ " 7 ! 22

Think: First the variable is by 3, and then 7 is .

To isolate the variable, subtract , and then multiply by .

$n3$ " 7 ! 22

# # Subtract to undo addition.

$n3$ !

! $n3$ ! ! 15 Multiply to undo division.

n !



Check

!n3! " 7 # 22

!!!!3" 7 ! 22 Substitute into the original equation.

" 7 ! 22 ✓

B. !y $

34

! # 9

Think: First is subtracted from the variable, and then the result is divided by

. To isolate the variable, by 3, and then 4.

!y $

34

! # 9

! !y $

34

! # ! 9 Multiply to undo division.

y $ 4 #

" " Add to undo subtraction.

y #

Check It Out!

1. The mechanic’s bill to repair your car was $850. The mechanic charges $35an hour for labor, and the parts that were used cost $275. How many hoursdid the mechanic work on your car?

2. Solve.

!n4! " 5 # 29

Chapter Review2

CHAPTER

2-1 Rational Numbers

Simplify.

1. !1236! 2. !

1383! 3. !

1546! 4. !

1360!

Write each decimal as a fraction in simplest form.

5. 0.45 6. 1.17 7. "3.04 8. 0.725

2-2 Comparing and Ordering Rational Numbers


9. !56! !

57! 10. "!

2115! !

1141! 11. !

183! !

2166! 12. 6.5 6!

35!

13. The lengths of four student’s pencils in Mr. Roberson’s class are 5.75 inches, 5.8 inches, 5!

56! inches, and 5.83 inches. List these measurements in order

from least to greatest.

2-3 Adding and Subtracting Rational Numbers


14. 74.28 " m for m # 31.72 15. "!23! $ t for t # !

73! 16. !

59! $ x for x # "3!

49!

17. Sammie and Lance both jog home after school every day. On Thursday, they made it home in 0.75 hour. Friday, it took them !

35! of an hour. How

much longer did it take Sammie and Lance to jog home on Thursday thanFriday?

2-4 Multiplying Rational Numbers


18. 2!37!!!

45!" 19. 3.4(6.7) 20. "!

14!!!

56!" 21. "7.2("9.1)

22. Kylie needs to cut 6 pieces of yarn that are each !23! foot long. How much

yarn does Kylie need?



2-5 Dividing Rational Numbers

Divide. Write each answer in simplest form.

23. !79! " !

23! 24. #!

27! " !

65! 25. !

34! " #2


26. !8h.6! for h $ 0.4 27. !

17p.8! for p $ 2 28. !

59.l22! for l $ 4.7

2-6 Adding and Subtracting with Unlike Demoninators

Add or Subtract.

29. !19! % !

45! 30. !

67! # !

13! 31. 3!

14! % 1!

16! 32. 5!

58! # 3!

12!

33. Sallie needs 4!14! cup of milk for a recipe. She has 2!

23! cups of milk. How

much more milk does Sallie need?

2-7 Solving Equations with Rational Numbers

Solve.

34. #7.1 # y $ 3.7 35. 62.3 % g $ 80.9 36. !4s.6! $ 11.4 37. w % !1

61! $ !

13!

38. Amy watched television for 2!34! hours. Charles watched television for !

13! of

the time Amy did. How long, in minutes, did Charles watch television?

2-8 Solving Two-Step Equations

Solve.

39. 3d % 7.3 $ 31 40. !c #

49

! $ #6 41. 12 % !2v

! $ 17 42. !13!u % !

34! $ !

16!

43. Gina bought a magazine subscription. The magazine company charged$15.75 for the subscription and $2.15 for each issue. If Gina paid $45.85,how many issues did she receive?

CHAPTER 2 REVIEW CONTINUED

Big Ideas2

CHAPTER

Answer these question to summarize the important concepts fromChapter 2 in your own words.

1. Explain how to write !2.54 as a fraction in simplest form.

2. Explain how to add "49" and "

29".

3. Explain how to divide rational numbers in fraction form.

4. Explain how to solve the equation x ! "15" # "1

45" .

For more review of Chapter 2:

• Complete the Chapter 2 Study Guide and Review on pages 110–112 ofyour textbook.

• Complete the Ready to Go On quizzes on pages 94 and 106 of yourtextbook.


Documents

LESSON Rational Numbers 2-1lfandres.weebly.com/uploads/2/5/8/0/25801520/pre_alg_2_chapter_notes.pdf" $ 10 3"1 1 0 " # " 3(10 1) 0 $ 1 " 10 10 # 10 Add numerators. Keep the denominator