Contents Lesson 6-1Ratios and Rates Lesson 6-2Using Proportions Lesson 6-3Scale Drawings and Models Lesson 6-4Fractions, Decimals, and Percents Lesson

Lesson 6-1 Ratios and Rates

Lesson 6-2 Using Proportions

Lesson 6-3 Scale Drawings and Models

Lesson 6-4 Fractions, Decimals, and Percents

Lesson 6-5 Using the Percent Proportion

Lesson 6-6 Finding Percents Mentally

Lesson 6-7 Using Percent Equations

Lesson 6-8 Percent of Change

Lesson 6-9 Probability and Predictions

Example 1 Write Ratios as Fractions

Example 2 Write Ratios as Fractions

Example 3 Find Unit Rate

Example 4 Convert Rates

Express the ratio 10 roses out of 12 flowers as a fraction in simplest form.

Divide the numerator and denominator by the GCF, 2.

The ratio of roses to flowers is 5 to 6. This means that for every 6 flowers, 5 of them are roses.

Answer:

Express the ratio 8 golden retrievers out of 12 dogs as a fraction in simplest form.

Answer:

Express the ratio 21 inches to 2 yards as a fraction in simplest form.

Convert 2 yards to inches.

Written in simplest form, the ratio is 7 to 24.

Answer:

Divide the numerator and denominator by the GCF, 3.

Express the ratio 4 feet to 18 inches as a fraction in simplest form.

Answer:

Shopping A 12-oz bottle of cleaner costs $4.50. A 16-oz bottle of cleaner costs $6.56. Which costs less per ounce?

Find and compare the unit rates of the bottles.

Divide the numerator and denominator by 12 to get a denominator of 1.

For the 12-oz bottle, the unit rate is $0.38 per ounce.

Divide the numerator and denominator by 16 to get a denominator of 1.

For the 16-oz bottle, the unit rate is $0.41 per ounce.

Answer: The 12-oz bottle has the lower cost per ounce.

Shopping A 6-pack of a soft drink costs $1.50. A 12-pack of a soft drink costs $2.76. Which pack costs less per can?

Answer: The 12-pack costs less per can.

Animals A snail moved 30 feet in 2 hours. How many inches per minute did the snail move?

You need to convert to .

There are 12 inches in 1 foot and 60 minutes in 1 hour.

Write 30 feet in 2 hours as .

.

Convert feet to inches and hours to minutes.

.

The reciprocal of

is .

.

Divide the common factors and units.

.6

1

1

2

3

1

Simplify.

Answer: 30 feet in 2 hours is equivalent to 3 inches per minute.

Jogging Dave jogs 2 miles in 22 minutes. How many feet per second is this?

Answer: This is 8 feet per second.

Example 1 Identify Proportions

Example 2 Solve Proportions

Example 3 Use a Proportion to Solve a Problem

Example 4 Convert Measurements

Determine whether the pair of ratios forms a proportion.

Write a proportion.

Cross products

Simplify.

Answer: So, .

Determine whether the pair of ratios forms a proportion.

Write a proportion.

Cross products

Simplify.

Answer: So, .

Determine whether each pair of ratios forms a proportion.

a.

b.

Answer:

Answer:

Solve the proportion .

Cross products

Multiply.

Divide.

Answer: The solution is 21.6.

Solve the proportion .

Cross products

Multiply.

Divide.

Answer: The solution is 5.

Solve each proportion.

a.

b.

Answer: 4.5

Answer: 7

Architecture An architect builds a model of a building before the actual building is built. The model is 8 inches tall and the actual building will be 22 feet tall. The model is 20 inches wide. Find the width of the actual building.

Explore You know the actual height of the building and the corresponding height of the model. You need to find the actual width of the building that corresponds with a model width of 20 inches.

Plan Write and solve a proportion using ratios that compare the actual building to the model. Let w represent the actual width of the building.

Write a proportion.

Cross products

Multiply.

Divide.

Simplify.

Solve

Answer: The actual width of the building is 55 feet.

Examine Check the cross products. Since and , the answer is correct.

Planes A model of a jet airplane has a length of 9 inches and a wingspan of 6 inches. Find the wingspan of the actual plane if the length is 120 feet.

Answer: The wingspan of the actual plane is 80 feet.

Attractions The Circleville Pumpkin Show in Circleville, Ohio, boasts the world’s largest pumpkin pie. The pie weighs 350 pounds and is 5 feet in diameter. Find the diameter of the pie in centimeters if 1 foot = 30.48 centimeters.

Let x represent the diameter in centimeters.customary measurement

metric measurement

customary measurement

metric measurement

Cross products

Simplify.

Answer: The diameter of the pie is 152.4 centimeters.

School The gymnasium in a new school building measures 55 feet in length. Find the length of the gymnasium in centimeters if 1 ft = 30.48 centimeters.

Answer: The length of the gymnasium is 1676.4 centimeters.

Example 1 Find Actual Measurements

Example 2 Determine the Scale

Example 3 Construct a Scale Drawing

Map A map has a scale of 1 inch = 8 miles. Two towns are 3.25 inches apart on the map. What is the actual distance between the two towns?

Let x represent the actual distance between the two towns. Write and solve a proportion.

map distance

actual distance

map distance

actual distance

Find the cross products.

Simplify.

Answer: The actual distance between the two towns is 26 miles.

Map A map has a scale of 1 inch = 8 miles. Two towns are 3.25 inches apart on the map. What is the scale factor?

To find the scale factor, write the ratio of 1 inch to 8 miles in simplest form.

Convert 8 miles to inches.

Answer: The scale factor is . That is, each

measurement on the map is

the actual measurement.

Scale Drawing A scale drawing of a new house has a scale of 1 inch = 4 feet. The height of the living room ceiling is 2.75 inches on the scale drawing.

a. What is the actual height of the ceiling?

b. What is the scale factor?

Answer: 11 feet

Answer:

Model Car A model car is 4 inches long. The actual car is 12 feet long. What is the scale of the model?

Write the ratio of the length of the model to the actuallength of the car. Then solve a proportion in which thelength of the model is 1 inch and the actual length is x feet.

model length

actual length

model length

actual length


Simplify.

Divide each side by 4.

Simplify.

Answer: The scale is 1 inch = 3 feet.

Log Cabin A model log cabin is 12 inches high. The actual log cabin is 42 feet high. What is the scale of the model?

Answer: The scale is 1 inch = 3.5 feet.

Patio Design Sheila is designing a patio that is 16 feet long and 14 feet wide. Make a scale drawing of the patio. Use a scale of 0.5 inch = 4 feet.

Step 1 Find the measure of the patio’s length on the drawing. Let x represent the length.

drawing length

actual length

drawing length

actual length


Simplify.


On the drawing, the length is 2 inches.

Step 2 Find the measure of the patio’s width on the drawing. Let w represent the width.

drawing width

actual width

drawing width

actual width


Simplify.


Simplify.

On the drawing, the width is 1.75 or inches.

Step 3 Make the scale drawing.

Use -inch grid paper. Since 2 inches =

8 squares and inches = 7 squares, draw

a rectangle that is 8 squares by 7 squares.

Answer:

Gardening Bob is designing a garden that is 18 feet long and 14 feet wide. Make a scale drawing of the garden. Use a scale of 0.5 inch = 4 feet.

Answer: The length is inches = 9 squares and the

width is inches = 7 squares.

Answer:

Example 1 Percents as Fractions

Example 2 Fractions as Percents

Example 3 Percents as Decimals

Example 4 Decimals as Percents

Example 5 Fractions as Percents

Example 6 Compare Numbers

Express 60% as a fraction in simplest form.

Answer:

Express 104% as a fraction in simplest form.

Answer:

Express 0.3% as a fraction in simplest form.

Answer:

Multiply by to eliminate

the decimal in the numerator.

The fraction bar indicates division.

Express as a fraction in simplest form.

Answer:

9

4

Express each percent as a fraction in simplest form.

a.

b.

c.

d.

Answer:

Answer:

Answer:

Answer:

Express as a percent.

Answer:

Answer:

Express as a percent.

Express each fraction as a percent.

a.

b.

Answer: 68%

Answer: 140%

Express 60% as a decimal.

Divide by 100 and remove the %.

Answer:



Answer:



Answer:

Express 0.4% as a decimal.


Answer:

Express each percent as a decimal.

a. 84%

b. 7%

c. 302%

d. 0.9%

Answer: 0.84

Answer: 0.07

Answer: 3.02

Answer: 0.009

Express 0.4 as a percent.

Answer:

Multiply by 100 and add the %.


Answer:



Answer:



Answer:


Express each decimal as a percent.

a. 0.84

b. 0.01

c. 0.004

d. 2.39

Answer: 84%

Answer: 1%

Answer: 0.4%

Answer: 239%

Express as a percent. Round to the nearest

tenth percent, if necessary.

Answer:



Answer:



Answer:



Answer:

Express each fraction as a percent. Round to the nearest tenth percent, if necessary.

a.

b.

c.

d.

Answer: 37.5%

Answer: 41.7%

Answer: 1.3%

Answer: 123.5%

Bakery A baker says that 25% of his customers buy

only bread and of his customers buy only

cookies. Which group is larger?

Write as a percent. Then compare.

Answer: Since 40% is greater than 25%, the group that buys only cookies is larger.

School The school principal states that of the

students are involved in instrumental music while

42% are involved in vocal music. Which group is

larger?

Answer: The group involved in vocal music is larger.

Example 1 Find the Percent


Example 3 Apply the Percent Proportion

Example 4 Find the Part

Example 5 Apply the Percent Proportion

Example 6 Find the Base

Twenty is what percent of 25?

Twenty is being compared to 25. So, 20 is the part and 25 is the base. Let p represent the percent.

Replace a with 20 and b with 25.


Simplify.


Simplify.

Answer: 20 is 80% of 25.

Twelve is what percent of 40?


What percent of 8 is 12?

Twelve is being compared to 8. So, 12 is the part and 8 is the base. Let p represent the percent.


Simplify.


Answer: 150% of 8 is 12.


Simplify.


What percent of 20 is 35?

Tennis From the years 1988 through 1996 Steffi Graf won the U.S. Open tennis championships five times and Wimbledon seven times. What percent of both tournaments combined during those years was Steffi Graf the women’s champion?

Compare the number of Steffi Graf’s wins, 12, to the total number of tournaments played, 18. Let a represent the part, 12, and let b represent the base, 18, in the percent proportion. Let p represent the percent.



Simplify.


Simplify.

Answer: Steffi Graf won about 66.7% of the tournaments.

Bake Sale At the school bake sale, 23 chocolate chip cookies, 18 oatmeal raisin cookies, and 7 peanut butter cookies were sold. If the sale started with a total of 90 cookies, what percent of the cookies were sold?

Answer: About 53.3% of the cookies were sold.

What number is 8.8% of 20?

The percent is 8.8, and the base is 20. Let a represent the part.

Answer: 8.8% of 20 is 1.76.

Replace b with 20 and p with 8.8.


Simplify.

Mentally divide each side by 100.

What number is 42.5% of 90?

Answer: 42.5% of 90 is 38.25.

Recycling At Kington Middle School, 600 students were asked if their families recycle cans and/or newspapers. 32% said they recycle both; 25% said they recycle cans; 19% said they recycle newspapers; 24% said they do not recycle. How many families recycle both cans and newspapers?

The total number of families is 600. So, 600 is the base. The percent is 32%.

To find 32% of 600, let b represent the base, 600, and let p represent the percent, 32%, in the percent proportion. Let a represent the part.

Replace b with 600and p with 32.


Simplify.


Answer: 192 families recycle both cans and newspapers.

Market Research A survey of 500 teenagers found that 37% prefer a soft drink with dinner; 48% prefer water; and 15% prefer milk. How many of the teenagers prefer milk with dinner?

Answer: 75 teenagers prefer milk with dinner.

Seventy is 28% of what number?

The percent is 28% and the part is 70. Let b represent the base.

Replace a with 70and p with 28.


Simplify.


Simplify.


Ninety is 24% of what number?


Example 1 Find Percent of a Number Mentally

Example 2 Estimate Percents

Example 3 Use Estimation to Solve a Problem

Find 50% of 46 mentally.

Think: .

Think: of 46 is 23.


Think: of 88 is 22.

Think: .



Think: .



Think: of 110 is 11.

So, of 110 is 77.

Find the percent of each number mentally.

a. 50% of 82

b. 25% of 36

c. 80% of 60

Answer: 41

Answer: 9

Answer: 48

Estimate 22% of 494.

22% is about 20% or .

494 is about 500.

of 500 is 100.

Answer: 22% of 494 is about 100.


63% is about 60% or .

788 is about 800.

of 800 is 480.


1% of 1200 is 12.

Estimate of 1219.

1219 is almost 1200.

of 1219 is about or 3.Answer:


155% means about 150 for every 100 or about 15 for every 10.

38 has about 4 tens.


a. Estimate 38% of 400.

b. Estimate 72% of 318.

c. Estimate of 2482.

d. Estimate 183% of 93.

Answer: 160

Answer: 210

Answer: 5

Answer: 162

Money A restaurant bill totals $21.35. You want to leave a 15% tip. What is a reasonable amount for the tip?

$21.35 is about $21.

So, 15% is about or $3.15.

Answer: A reasonable amount for the tip would be $3.

10% of $21 is $2.10. Move the decimal point1 place to the left.

5% of $21 is $1.05. 5% is one half of 10%.

Money A restaurant bill totals $59.05. You want to leave a 15% tip. What is a reasonable amount for the tip?

Answer: A reasonable amount for the tip would be $9.

Example 1 Find the Part


Example 3 Find the Base

Example 4 Find Discount

Example 5 Apply Simple Interest Formula

Find 38% of 22. Estimate: 40% of 20 is 8.

You know that that the base is 22 and the percent is 38%. Let n represent the part.

Write 38% as the decimal 0.38.

Simplify.

Answer: 38% of 22 is 8.36.

Find 64% of 48.

Answer: 64% of 48 is 30.72.

19 is what percent of 25?

You know that the base is 25 and the part is 19. Let n represent the percent.


Estimate:

, which is 80%.


Simplify.

The answer makes sense compared to the estimate.

8 is what percent of 25?


84 is 16% of what number? Estimate: 80 is 16% of 500.

You know that the part is 84 and the percent is 16%. Let n represent the base.

Divide each side by 0.16.

Write 16% as the decimal 0.16.


Simplify.

The answer is reasonable since it is close to the estimate.

315 is 42% of what number?

Answer: 315 is 42% of 750.

Jewelry The regular price of a ring is $495. It is on sale at a 20% discount. What is the sale price of the ring?

Method 1First, use the percent equation to find 20% of 495.

Estimate:

Let d represent the discount.

The base is 495 and the percent is 20%.

Simplify.

Then, find the sale price.

Subtract the discount from the original price.

Method 2

Let s represent the sale price.

A discount of 20% means the ring will costor 80% of the original price. Use the percent equation to find 80% of 495.

The base is 495 and the percent is 80%.

Answer: The sale price of the ring will be $396.

Retail The regular price of a stereo system is $1295. The system is on sale at a 15% discount. Find the sale price of the stereo system.

Answer: The sale price of the stereo system is $1100.75.

Banking Suppose you invest $2000 at an annual interest rate of 4.5%. How long will it take for it to earn $495?

Write the simple interest formula.

Replace I with 495, p with 2000, and r with 0.045.

Simplify.


Simplify.

Answer: It will take 5.5 years to earn $495.

Banking Suppose you invest $3500 at an annual interest rate of 6.25%. How long will it take for it to earn $875?

Answer: It will take 4 years to earn $875.

Example 1 Find Percent of Change

Example 2 Find Percent of Increase

Example 3 Find Percent of Increase

Example 4 Find Percent of Decrease

Find the percent of change from 325 to 390.

Step 1 Subtract to find the amount of change.

Step 2 Write a ratio that compares the amount of change to the original amount. Express the ratio as a percent.

Substitution.

Write the decimal as a percent.

Answer: The percent of change from 325 to 390 is 20%.

Find the percent of change from 84 to 105.

Answer: The percent of change from 84 to 105 is 25%.

Tuition In 1965, when John entered college, the tuition per year was $7500. In 2000, when his daughter went to the same school, the tuition was $25,500. Find the percent of change.


Step 2 Write a ratio that compares the amount of change to the original tuition. Express the ratio as a percent.

Answer: The percent of change is 240%. In this case, the percent of change is a percent of increase.

Substitution.


Textbooks In 1990, the price of a textbook was $38. In 2000, the price of the same textbook was $81. Find the percent of change.

Answer: The percent of change was about 113%. In this case, the percent of change is a percent of increase.

Multiple-Choice Test Item

Refer to the table shown. Which county had the least percent of increase in population from 1990 to 2000?

County 1990 2000

Breckenridge 16,312 18,648

Bracken 7766 8279

Calloway 30,735 34,177

Fulton 8271 7752

A Breckinridge B BrackenC Calloway D Fulton

Read the Test Item Percent of increase tells how much the population has increased in relation to 1990.

Solve the Test ItemUse a ratio to find each percent of increase. Then compare the percents.

• Breckinridge

• Bracken

• Calloway

• Fulton

Answer: Bracken County had the least percent of increase in population from 1990 to 2000. The answer is B.

Eliminate this choice because the population decreased.

Multiple-Choice Test Item

The table shows test scores on the first two math tests of the semester for four students. Which student had the greatest percent of increase from test 1 to test 2?

A Holly B Ben C Sally D Max

Student Name Test 1 Score Test 2 Score

Holly 84 91

Ben 93 89

Sally 65 79

Max 73 98

Answer: Max had the greatest percent of increase from test 1 to test 2. The answer is D.

Clothing A $110 sweater is on sale for $88. What is the percent of change?

Step 2 Compare the amount of change to the original price.


Substitution.


Answer: The percent of change is –20%. In this case, the percent of change is a percent of decrease.

Shoes A $145 pair of tennis shoes is on sale for $105. What is the percent of change?

Answer: The percent of change is about –27.6%.

Example 1 Find Probability

Example 2 Find Probability

Example 3 Find Experimental Probability

Example 4 Make a Prediction

Suppose a number cube is rolled. What is the probability of rolling a 4 or a 5?

There are 2 numbers on the number cube that are a 4 or a 5: 4 and 5.

There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.

Answer: The probability of rolling a 4 or a 5 is

Suppose a number cube is rolled. What is the probability of rolling a number that is divisible by 3?

Answer: The probability of rolling a number that is

divisible by 3 is

Suppose that two number cubes are rolled. Find the probability of rolling two identical numbers.

Make a table showing the sample space when rolling two number cubes.

1 2 3 4 5 6

1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)

3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)

4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)

5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)

6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

There are 6 outcomes in which the two numbers are identical.

Answer:

This means there is a

chance of rolling two identical numbers.

Suppose that two number cubes are rolled. Find the probability of rolling two numbers whose sum is 8.

Answer: There is a chance of

rolling two numbers whose sum is 8.

Use the table below to determine the experimental probability of landing on heads for this experiment.

Answer: The experimental probability of landing on

heads in this case is

Basketball Brian is shooting baskets with a basketball. He makes 13 shots and misses 9 shots. Determine the experimental probability of Brian making a shot.

Answer: The experimental probability of making

a shot in this case is

Sports Miss Newman surveyed her class to see which sport they preferred watching. 44% preferred football, 28% basketball, 20% soccer, and 8% tennis. Out of 560 students in the entire school, how many would you expect to say they prefer watching basketball?

The total number of students is 560. So, 560 is the base. The percent is 28%.

To find 28% of 560, let b represent the base, 560, and let p represent the percent, 28%, in the percent proportion. Let a represent the part.

part

basepercent

Simplify.


Answer: You can expect about 157 students to say they prefer watching basketball.

Colors The students in an art class were surveyed about their favorite color. 32% preferred blue, 29% preferred red, 23% preferred yellow, and 16% preferred green. Out of 450 students in the entire school, how many would you expect to say they prefer red?

Answer: You can expect about 131 students to say they prefer red.

Documents

Contents Lesson 6-1Ratios and Rates Lesson 6-2Using Proportions Lesson 6-3Scale Drawings and Models Lesson 6-4Fractions, Decimals, and Percents Lesson