Embed Size (px)
Citation preview
Unit 2Rational Numbers
Section 1Factors and Prime Factorization
Yearbook Problem:You are working on your school yearbook. Each page will have 24 student photos. The photos will be arranged in a rectangular display with the same number of photos in each row. How many ways can you arrange the photos so that there are no more than 10 photos in any row or column?
Factors: The whole numbers that divide the given number _______ ____________________________________
The factors of 24 are:
Example 1.1Find all the factors of the number:a. 30
b. 31
c. 45
d. 87
Prime Number: a whole number that is greater than 1 and has exactly two whole number factors, ____ and _____________
Composite Number: a whole number that is greater than 1 and has _______________________________whole number factorsThe number 1 is neither prime nor composite
Prime Factorization: Rewriting a number as a product of ___________________________________
Factor Tree: A diagram to assist writing the _________________ ______________________________________________________
Examples of Prime and Composite Numbers
Example 1.2Use a factor tree to write the prime factorization of 630.
Example 1.3Tell whether each number is prime or composite. If it is composite, write its prime factorization.
a. 32
b. 56
c. 59
d. 83
e. 101
f.175
g. 180
h. 420
MonomialsMonomial: a number, a variable, or the product of a ___________ and one or more ______________ raised to whole number powers.
Factoring a MonomialTo factor a monomial, write it as a product of prime numbers and variables with exponents of 1.
Example 1.4Factor: 28xy3
Example 1.5Factor:1. 6ab
2. 15n3
3. 3x3y2
4. 36s4t
Section 2Greatest Common Factor
Choir:Mr. Donner wants to divide the choir into smaller groups. The choir has 24 sopranos, 60 altos, and 36 tenors. Each group will have the same number of each type of voice. What is the greatest number of groups that can be formed? How many sopranos, altos, and tenors will be in each group?
Common Factor: A whole number that is a factor of two or more ____________________________Greatest Common Factor (GCF): The greatest of the _________ ______________________
Example 2.1Find the greatest common factor of the numbers.a. 12, 30
b. 21, 42
c. 16, 32, 40
d. 27, 45, 90
Relatively Prime: Two or more numbers with a greatest common factor of ______
Example 2.2Find the GCF of the numbers. Then tell whether the numbers are relatively prime.
a. 24, 45
b. 35, 54
Example 2.3Find the GCF of the numbers. Then tell whether the numbers are relatively prime.a. 18, 33
b. 39, 50
c. 110, 77
d. 21, 160
GCF of MonomialsFind the GCF by factoring each monomial!
Find the GCF of 18xy2 and 28x2y2
Example 2.4Find the GCF.a. 6x, 15x
b. 20x2, 36x
c. 32y2, 6x2y
d. 7xy3, 28xy2
Section 3Ordering and Comparing Fractions
Equivalent FractionsTwo fractions that represent the same number.
To write equivalent fractions, multiply or divide the numerator and denominator by the same nonzero number.
Example 3.1Write two fractions that are equivalent to 8
12
On Your OwnWrite two fractions that are equivalent to the given fraction.
a. 510
b. 69
c. 1220
d. 1824
Simplest FormA fraction is in simplest form when its numerator and denominator are _____________________________
To find simplest form: divide the numerator and denominator by their _________________
Example 3.2Write 12
30 in simplest form.
Example 3.3Write the fractions in simplest form:
a. 414
b. 836
c. 2742
d. 2849
Variable ExpressionsTo simplify fractions that contain variables, factor the numerator and the denominator. Then _______________________________
Example 3.4Write in simplest form:
10xy15 y2
Section 4Least Common Multiple
AgricultureCrop rotation is a system in which farmers vary the crops they plant in their fields each year. Suppose a farmer grows alfalfa in a certain field every 6 years. In another field, the farmer grows alfalfa every 10 years. This year, the farmer is growing alfalfa in both fields. In how many years will the farmer grow alfalfa in both fields again?
Multiple: The product of the number and ____________________ ______________________Common Multiple: A multiple that is shared by ______________ ________________________Least Common Multiple (LCM): The smallest of the ________ _______________ of two or more numbers
Example 4.1Find the multiples of 8, the multiples of 12, the common multiples of 8 and 12, and the LCM of 8 and 12.
Example 4.2Find the least common multiple of the numbers:a. 16 and 24
b. 20 and 25
c. 6, 8, and 20
d. 15, 30, and 50
Example 4.3Find the least common multiple of 9xy2 and 15x2y.
Example 4.4Find the least common multiple of the monomials:1. 15x2 and 27x
2. 6m2 and 10m3
3. 14ab and 21bc
4. r2 and 5rst
Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the __________________________________________________. You can use LCD to compare and order fractions.
Example 4.5Use the LCD to determine which fraction is greater.
1. 56 and 7
9
2. 58 and 13
20
3. 712 and 11
15
4. 516 and 3
10
Example 4.6Order the numbers 3 4
15 , 3310, and 19
6 from least to greatest.
Section 5Rational Numbers
Rational Number: a number that can be written as a ___________ ______________________
Example 5.1Are the following numbers rational (can they be written as a quotient of two integers)?a. 7
b. -10
c. 5 ¾
d. -3 ½
Terminating Decimal: when a rational number is divided, the quotient ends and you obtain a _____________________________Repeating Decimal: when the division is carried out, a digit or block of digits in the quotient _____________________________
Example 5.2Write the following numbers as decimals:a. 3
8
b. 511
Example 5.3Write the following numbers as decimals:a. 7
12
b. 27
Example 5.4Of the 50 mammal species found in Canyonlands National Park, 20 species belong to the order Rodentia. Of the 54 mammal species found in Badlands National Park, 24 belong to Rodentia. In which park is the fraction of mammal species belonging to Rodentia greater?
With a PartnerWrite the fraction or mixed number as a decimal:a. 3
10
b. −23
c. 1 920
d. 2980
Writing a Decimal as a FractionTerminating Decimals: use the place of the last digit to determine the _______________________________________________
Example 5.5Write the number as a fraction:A. 0.7 B. -3.05
Example 5.6Write the number as a fraction:A. 5.54 B. -0.895
Repeating Decimal: set x equal to the decimal. Multiply each side of the equation by 10n where n is the number of repeating digits in the decimal. Subtract x from both sides of the equation and divide.
Example 5.7Write 0.93 as a fraction.
Example 5.8Write as a fraction:A. 0.8 B. 2.707
Critical ThinkingCompare writing 0.3 as a fraction with writing 0.3 as a fraction. Discuss with a partner.
Section 6Adding and Subtracting Like Fractions
Like Fractions: fractions with the same _____________________
Adding and Subtracting Like FractionsWrite the sum or difference of the numerators over the denominator
Example 6.149+ 1
9
911
− 211
Example 6.2One night, 77/100 of the moon's visible surface is illuminated. The next night, an additional 9/100 is illuminated. What fraction of the moon's visible surface is illuminated on the second night?
Example 6.3−47
−27
110
−(−310
)
Example 6.438+ 2
8
−16
+ 56
Example 6.52
15− 7
15
112
−(−712
)
Mixed Numbers
First write the mixed numbers as improper fractions, then add or subtract as before.
Example 6.65 5
9+2 7
9
−10 613
−6 813
Example 6.72 3
4+1 3
4
−6 23+3 1
3
Example 6.84 1
5−2 3
5
−3 27−6 3
7
Simplifying Variable Expressions3a20
+ 5a20
−83b
−(−23b
)
Section 7Adding and Subtracting Unlike Fractions
Step 1: Rewrite the fractions using the LCD.
Step 2: Add or subtract the numerators and put this value over the denominator.Step 3: Simplify.
Example 7.1512
+ 13
−56
−79
Example 7.2−4 2
5+(−2 6
11) −2
3+ 1
4
Example 7.33
10−4
5
−415
− 910
Example 7.43 5
9+2 1
66 7
10+(−1 1
5)
Example 7.5−2 1
3+6 3
5−14
−(−23
)
Example 7.6You are hiking between two campsites in the Chesapeake and Ohio Canal National Historic Park. The distance between the campsites is 10 1
5 miles. You have already hiked 5 3
4 miles. How many more miles do you have to hike?
Example 7.75 4
11−2 2
3−1 3
7−2 3
14
Example 7.84 3
8−(−1 2
3) −3
8+ 1
2
Simplifying an Expressiona2−a
6m5
−2m3
Example 7.9x4−5 x
6−5w
12+7w
9
Section 8Multiplying Fractions
The product of two or more fractions is equal to the product of the numerators over the product of the denominators.
35• 4
7=3•4
5 •7=12
35
Example 8.1
710•(−4
21) 2
3• 7
8
Example 8.2(−5
12)( 3
10) 10 •(−2 3
11)
Example 8.3( 34)(−12) −3
5• 11
12
Example 8.4Each year, the Academy of Television Arts and Sciences presents gold-plated Emmy awards for programs and individuals in the television industry. Each Emmy statue
weighs 4 34 pounds. In 2002, 63 statues were awarded. What
was the combined weight of all the statues?
Example 8.5−2 3
4•3 1
54 7
8•5 2
3
Example 8.6−3 2
7•1 1
2−3 3
5•(−1 5
9)
Example 8.74 1
8•(−1 2
3)
7 23•2 7
10
Example 8.8m3•(−12
5) n2
10• 5n3
9
Example 8.93 y4• y
5
9x4
6•(−16 x
3)
Example 8.103 z5
25• 2 z4
15
−4 v2
21• 7v3
16
Section 9Dividing Fractions
Steps
1.
2.
3.
4.5.
311÷ 3
432÷ 11
223÷ 10
7
Divide and Simplify34÷ 2
345÷ 1
247÷ 1
3
13÷ 1
357÷ 2
389÷ 3
5
13÷ 3
8
36÷ 2
3
Dividing Mixed Numbers3 1
3÷1 1
25 2
5÷2 2
34 2
4÷2 4
6
Divide:1.
2.
3.
4.
5.
Section 10Ratios and Rates
Think About ThisAn archer shoots 60 arrows at a target, with 44 arrows hitting the scoring area and 16 missing the scoring area. How can you evaluate the archer's performance?
Writing RatiosA ratio uses division to ________________________________
Ratios of two quantities (a and b) can be written in three ways:1.
2.
3.
Each of these three ratios is read as "the ratio of a to b"
Ratios should be written in simplest form
Think back to the archer we talked about (with 60 arrows and hitting the scoring area with 44 of them). We can write two different ratios, each in three different ways.a. The number of hits to the number of misses.
b. The number of hits to the number of shots.
With the people in your pod, write the ratio of the number of misses to the number of shots using a ratio. Write this ratio in three different ways.
Equivalent Ratios: have the same value when ________________
Comparing RatiosSteps to compare ratios:
1. Put all ratios in a common form (pick whichever you like)2. Get all ratios into common terms (for example common
denominators)3. Compare and order ratios
Example 10.1Put the following ratios in order from least to greatest:
42,11¿2,22:3 , 30
4,36 :5
RatesA rate is a ratio of two quantities measured in ________________
A unit rate is a rate that has a denominator of ________ when expressed in fraction form
Often expressed using the word "per" which means "for every"
Example 10.2You host a party for 12 people. The food and drinks for the party cost $66. What is the cost per person?
Example 10.3A jet flies 540 miles per hour. Write its rate in miles per minute.
Example 10.4Engineers designed a miniature robot that can crawl through pipes and vents that humans can't access. The robot travels 1 inch in 3 seconds. How many feet does the robot travel in 4 minutes?
With Your PartnerWrite the equivalent rate:
a. 5cm1min
=¿¿cm
1hour
b. 5m1 sec
= ¿¿
1hour
Section 11Writing and Solving Proportions
Each day, an elephant eats 5 pounds of food for every 100 pounds of its body weight. How much does a 9300 pound elephant eat per day?
Proportion: an equation that states that two ratios _____________
Solving a ProportionIf one of the numbers in a proportion is unknown, you can solve the proportion for that unknown number. There are three methods to solve proportions:1.
2.
3.
Example 11.1Solve the proportion:
56= x
18
Example 11.2Solve the proportion:
27= x
21
Try With a Neighbor38= x
32x2=20
10x
48= 6
12
Example 11.3Solve the proportion:
x12
=28
Example 11.425= x
25With Your Neighbor
310
= x100
x9=42
54x4=13
2
Setting Up a ProportionThere are different ways to set up a proportion. Solve the following problem using two different proportions. Yesterday you rode your bike 18 miles in 2.5 hours. Today you plan to ride for 3.5 hours. If you ride at the same rate as yesterday, how far will you ride?
Think back to the elephant problem at the beginning of the hour. If the elephant eats 5 pounds of food for every 100 pounds of its body weight, how much does a 9300 pound elephant eat per day? Solve this problem using proportions.
Section 12Solving Proportions Using Cross Products
Cross ProductsCross Product of Two Ratios: the product of the _____________ of one ratio and the __________________ of the other ratio. Every set of ratios has _________ cross products.
If the cross products are equal, then the ratios are equal and vise versa.
Example 12.1Tell whether the ratios form a proportion:a. 9
51 and 634 b. 12
20 and 3250
With your NeighborTell whether the ratios form a proportion:
a. 614 and 3
7
b. 1435 and 8
20
c. 611 and 9
16
d. 1524 and 10
16
Cross Products Property: in a proportion, the product of the means is equal to the ____________________________________
In the proportion above: ad=bc
Example 12.2Solve the proportion:
1842
=3t
Example 12.316p
=1045
With Your Neighbor9b=1.5
70.46
=18z
Example 12.4Human hair grows about 0.7 centimeter in 2 weeks. How long does hair take to grow 14 centimeters?
Solve the Proportion Using All Three Methods512
= x36
Section 13Percents
Percent: a ratio whose denominator is __________Percent= per 100 (for every 100)
Example 13.1Write 29% and 45% as fractions in simplest form.
Example 13.2Write 7
10 and 35 as percents.
With Your NeighborWrite 51% and 25% as fractions in simplest form.
Write 11/20 and 4/25 as percents.
Common Percents as Fractions
Example 13.3
A desert tortoise can go a year or more without drinking water. When it does drink, its body weight can increase by 40%. Suppose a desert tortoise weighs 15 pounds after a long period without water. How many pounds can the tortoise gain when it drinks?
Example 13.4Find the percent of the number:25% of 36
70% of 70
With Your NeighborFind the percent of the number:50% of 14
75% of 80
Section 14Percents and Proportions
Solving Percent ProblemsYou can represent a percent using the proportion:
where a is part of b and p% is the percent that a is of b.
Example 14.1What percent of 7 is 4?
Example 14.2What percent of 72 is 54?
With Your Neighbora. What percent of 60 is 25?
b. What percent of 90 is 40? c. What percent of 35 is 7?
Example 14.3What number is 24% of 200?
Example 14.4836 is what 16% of what number?
Example 14.5What number is 18% of 50?
Example 14.6105 is 84% of what number?
With your NeighborSolve:What number is 30% of 130?
48 is 75% of what number?
11 is 22% of what number?
What number is 55% of 80?
Three Types of Percent Problems
Section 15
Percents and Decimals
The African pygmy hippo is the smallest species of hippopotamus. Suppose a common adult hippo weighs 5600 pounds, and an adult pygmy hippo's weight is 10.5% of the common adult hippo's weight. How much does the adult pygmy hippo weigh?
•To write a decimal as a percent, move the decimal point two places to the right and write a percent sign.
•To write a percent as a decimal, move the decimal point two places to the left and remove the percent sign.
Example 15.1Write 0.62, 1, and 2.3 as percents.
Example 15.2Write 75%, 0.4%, and 168% as decimals.
With Your NeighborWrite the decimal as a percent:a. 0.461 b. 5 c. 1.9 d. 0.007
Write the percent as a decimal:e. 27% f. 184% g. 3% h. 0.55%
FractionsA fraction, decimal, and percent can all represent the same number. We wrote fractions as percents, but only fractions with denominators that have 100 as a multiple. What happens if the denominator is not a factor of 100?
Example 15.3Write 3
8 as a percent.
Example 15.4Write 5
3 as a percent.With Your NeighborWrite the fraction as a percent.a. 7
8
b. 512
c. 116
d. 54
Hippos!Find the weight of the pygmy hippo described before (10.5% of the weight of a common hippo. Common hippo weighs 5600 pounds).
Section 16The Percent Equation
The Percent EquationYou can represent "a is p percent of b" using the equation:
where a is a part of the base b and p% is the percent.
Example 16.1What number is 16% of 75?
Example 16.2What number is 89% of 110?
Example 16.3On June 14, 2002, the distance between Earth and the moon was about 375,000 kilometers. On that day, a traveling asteroid missed Earth by about 32% of that distance. How far away from Earth was the asteroid at that time?
Example 16.4Commission: A salesperson earns a 6.5% commission on every car sold. The salesperson sells a car for $21,800. What is the commission?
With a NeighborWhat number is 60% of 25?
What number is 20% of 45?
What number is 150% of 96?
Example 16.5What percent of 25 is 60?
Example 16.6What percent of 48 is 45?
Example 16.727 is 7.5% of what number?
Example 16.818 is 36% of what number?
Example 16.9Your friend paid $9 for a movie ticket. This amount was 72% of the total amount your friend spent at the theater. How much money did your friend spend?
With a NeighborWhat percent of 48 is 36?
What percent of 25 is 24?
9 is 0.03% of what number?
90 is 120% of what number?
SummaryTo find the percent of a number:
20% of 35 = 1/5 • 35 = 75% of 16 = 0.05 • 16 = 0.8
To find the percent p%, the base b, or a part of the base:Use proportions: a/b=p/100Use the percent equation: a = p% • b
Section 17Percent of Change
Percent of ChangeThe ratio of the amount of increase or decrease to the __________ ________________
Percent of change, p%=
Percent of Change: how much a quantity increases or decreases with respect to the original amountPercent of Increase: the percent of change if the new amount is greater than the original amountPercent of Decrease: the percent of change if the new amount is less than the original amount
Finding a Percent of IncreaseOriginal: 20New: 25
Example 17.1Original: 150New: 189
Example 17.2Original: 55New: 143
Example 17.3The International Balloon Fiesta takes place every year in Albuquerque, New Mexico. In 1999, 903 balloons participated. In 2000, 1019 balloons participated. By about what percent did the number of balloons increase from 1999 to 2000?
Finding a Percent of DecreaseOriginal: 512New: 320
Example 17.4Original: 20New: 15
Example 17.5Original: 75New: 35
Example 17.6Original: 102New: 51
Steps to Calculate Percent of Change1. Determine whether it is an increase or decrease.
2. Subtract the smaller number from the larger number.3. Put the answer from step 2 in the numerator of your fraction.4. Put the original number in the denominator.5. Divide the fraction to get a decimal.6. Convert the decimal to a percent.
Finding a New AmountIf you know the original amount and the percent of change, you can find the new amount.1. Multiply the percent of change by the original amount to find the amount of change.2. Increase or decrease the original amount by the amount of change.Example 17.7Increase 45 by 20%
Example 17.8Decrease 85 by 28%
Example 17.9There were about 198,000 spectators at an action sports event in 1995. The number of spectators increased by about 12% from 1995 to 2002. About how many spectators were there in 2002?
Another Way to Find a New AmountFor a p% increase: multiply the original amount by
For a p% decrease: multiply the original amount by
Example 17.10In 1983, the average price of an audio CD was $21.50. By 2000, the average price had decreased by 34.8%. What was the average price of a CD in 2000?
Example 17.11Increase 25 by 24%
Example 17.12Increase 120 by 75%
Example 17.13Decrease 35 by 60%
Example 17.14Decrease 72 by 65%
Section 18Percent Applications
MarkupsA retailer buys items from manufacturers at __________________. The retailer then sells those items to customers at _____________. An increase from the wholesale price of an item to the retail price is a __________________. The markup is calculated using a percent of the _______________________.
Example 18.1A street vendor buys bracelets from a manufacturer for $7 each. The vendor marks up the price by 150%. What is the retail price?
Example 18.2What would be the price of the bracelet in the last example if the markup percent is 120%?
DiscountA _________________ from the original price of an item to the sale price is a discount. The discount is calculated using a percent of the _____________________________Example 18.3You buy an electronic organizer that is on sale for 15% off the original price of $25. What is the sale price?
Example 18.4The bill for your restaurant meal is $22. You leave a 15% tip. The sales tax is 6%. What is the total cost of your meal?
Example 18.5A pair of jeans that originally costs $42 is 25% off. Find the sale price.
Example 18.6A furniture store marks up the wholesale price of a desk lamp by 80%. The retail price is $35. What is the wholesale price?
Example 18.7A store marks up the wholesale price of a printer by 80%. The retail price is $120. What is the wholesale price of the printer?
Section 19Simple and Compound Interest
Interest: the amount earned or paid for the ___________________Principal: the amount of money ___________________________Simple Interest: interest that is earned or paid only on the ____________________Annual Interest Rate: the percent of the principal earned or paid _____________________Balance: the sum of the ______________ and the _____________Compound Interest: interest that is earned on both the _________ and any _________________ that has been earned previously
Simple Interest FormulaI = P r tI is the _______________________
P is the __________________r is the ______________________________ (written as a decimal)t is the ________________________
Example 19.1Find the interest earned after 2 years for a bond that cost $1500 and earns 4% simple interest per year.
Example 19.2A $1000 bond earns 6% simple annual interest. What is the interest earned after 4 years?
Finding BalancesA = P + Prt or A = P(1 + rt)A is the _______________P is the _______________Prt is the _______________
Example 19.4You deposit $1000 in a savings account that earns 2.5% simple interest yearly. How much money is in the account after 2 years?
Example 19.4
You take out a loan for $2200 with a rate of 7% simple interest yearly. What is the total amount owed after 8 years assuming you haven't paid anything back?
Example 19.5You get a summer job at a bakery. Suppose you save $1400 of your pay and deposit it into an account that earns simple annual interest. After 9 months, the balance is $1421. Find the annual interest rate.
Example 19.6If the balance of your savings account is $1424.50 after 6 months and the account has a simple interest rate of 3.5%, what was the principal amount put into the account?
Compound Interest
Suppose you deposit $50 into a savings account that earns 2% interest compounded annually. Let's make a table of the balance after each of 3 years.
Compound Interest FormulaIf interest is compounded annually,A = P(1 + r)t
A is the _______________P is the _______________r is the ________________________ (written as a decimal)t is the _______________________________
Example 19.7You deposit $1500 into an account that earns 2.4% interest compounded annually. Find the balance after 6 years.
Example 19.8You deposit $680 into an account that earns 6.2% interest compounded annually. Find the total in the account after 10 years.
Example 19.9You take out a loan for a repair to your snowmobile for $1750 at an interest rate of 2.3% compounded annually. Assuming you don't make any payments, what is the total that you owe after 4 years?
Example 19.10
You deposit $1400 into an account that earns 4% interest compounded annually. What will the balance of the account be after 5 years?
