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Prealgebra 2e

5.6 Ratios and Rate

Prealgebra 2e5.6 Ratios and Rate

Learning Objectives

By the end of this section, you will be able to:

  • Write a ratio as a fraction
  • Write a rate as a fraction
  • Find unit rates
  • Find unit price
  • Translate phrases to expressions with fractions

Be Prepared 5.16

Before you get started, take this readiness quiz.

Simplify: 1624.
If you missed this problem, review Example 4.19.

Be Prepared 5.17

Divide: 2.76÷11.5.
If you missed this problem, review Example 5.19.

Be Prepared 5.18

Simplify: 112234.
If you missed this problem, review Example 4.43.

Write a Ratio as a Fraction

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare a and b, the ratio is written as atob,ab,ora:b.

Ratios

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of a to b is written atob,ab,ora:b.

In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as 41 instead of simplifying it to 4 so that we can see the two parts of the ratio.

Example 5.58

Write each ratio as a fraction: 15to27 45to18.

Try It 5.115

Write each ratio as a fraction: 21to56 48to32.

Try It 5.116

Write each ratio as a fraction: 27to72 51to34.

Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio 0.8to0.05. We can write it as a fraction with decimals and then multiply the numerator and denominator by 100 to eliminate the decimals.

A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.

Do you see a shortcut to find the equivalent fraction? Notice that 0.8=810 and 0.05=5100. The least common denominator of 810 and 5100 is 100. By multiplying the numerator and denominator of 0.80.05 by 100, we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:

The top line says 0.80 over 0.05. There are blue arrows moving the decimal points over 2 places to the right.
"Move" the decimal 2 places. 805
Simplify. 161

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

Example 5.59

Write each ratio as a fraction of whole numbers:

  1. 4.8to11.2
  2. 2.7to0.54

Try It 5.117

Write each ratio as a fraction: 4.6to11.5 2.3to0.69.

Try It 5.118

Write each ratio as a fraction: 3.4to15.3 3.4to0.68.

Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

Example 5.60

Write the ratio of 114to238 as a fraction.

Try It 5.119

Write each ratio as a fraction: 134to258.

Try It 5.120

Write each ratio as a fraction: 118to234.

Applications of Ratios

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than 5 to 1 is considered good.

Example 5.61

Hector's total cholesterol is 249 mg/dl and his HDL cholesterol is 39 mg/dl. Find the ratio of his total cholesterol to his HDL cholesterol. Assuming that a ratio less than 5 to 1 is considered good, what would you suggest to Hector?

Try It 5.121

Find the patient's ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is 185 mg/dL and HDL cholesterol is 40 mg/dL.

Try It 5.122

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is 204 mg/dL and HDL cholesterol is 38 mg/dL.

Ratios of Two Measurements in Different Units

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.

We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

Example 5.62

The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of 1 inch for every 1 foot of horizontal run. What is the ratio of the rise to the run?

Try It 5.123

Find the ratio of the first length to the second length: 32 inches to 1 foot.

Try It 5.124

Find the ratio of the first length to the second length: 1 foot to 54 inches.

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are 120 miles in 2 hours, 160 words in 4 minutes, and $5 dollars per 64 ounces.

Rate

A rate compares two quantities of different units. A rate is usually written as a fraction.

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

Example 5.63

Bob drove his car 525 miles in 9 hours. Write this rate as a fraction.

Try It 5.125

Write the rate as a fraction: 492 miles in 8 hours.

Try It 5.126

Write the rate as a fraction: 242 miles in 6 hours.

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of 175 miles3 hours. This tells us that every three hours, Bob will travel 175 miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of 1 unit is referred to as a unit rate.

Unit Rate

A unit rate is a rate with denominator of 1 unit.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of 68 miles per hour we mean that we travel 68 miles in 1 hour. We would write this rate as 68 miles/hour (read 68 miles per hour). The common abbreviation for this is 68 mph. Note that when no number is written before a unit, it is assumed to be 1.

So 68 miles/hour really means 68 miles/1 hour.

Two rates we often use when driving can be written in different forms, as shown:

Example Rate Write Abbreviate Read
68 miles in 1 hour 68 miles1 hour 68 miles/hour 68 mph 68 miles per hour
36 miles to 1 gallon 36 miles1 gallon 36 miles/gallon 36 mpg 36 miles per gallon

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $12.50 for each hour you work, you could write that your hourly (unit) pay rate is $12.50/hour (read $12.50 per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of 1.

Example 5.64

Anita was paid $384 last week for working 32 hours. What is Anita’s hourly pay rate?

Try It 5.127

Find the unit rate: $630 for 35 hours.

Try It 5.128

Find the unit rate: $684 for 36 hours.

Example 5.65

Sven drives his car 455 miles, using 14 gallons of gasoline. How many miles per gallon does his car get?

Try It 5.129

Find the unit rate: 423 miles to 18 gallons of gas.

Try It 5.130

Find the unit rate: 406 miles to 14.5 gallons of gas.

Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Unit price

A unit price is a unit rate that gives the price of one item.

Example 5.66

The grocery store charges $3.99 for a case of 24 bottles of water. What is the unit price?

Try It 5.131

Find the unit price. Round your answer to the nearest cent if necessary.

24-pack of juice boxes for $6.99

Try It 5.132

Find the unit price. Round your answer to the nearest cent if necessary.

24-pack of bottles of ice tea for $12.72

Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

Example 5.67

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $14.99 for 64 loads of laundry and the same brand of powder detergent is priced at $15.99 for 80 loads.

Which detergent has the lowest cost per load?

Try It 5.133

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand A Storage Bags, $4.59 for 40 count, or Brand B Storage Bags, $3.99 for 30 count

Try It 5.134

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand C Chicken Noodle Soup, $1.89 for 26 ounces, or Brand D Chicken Noodle Soup, $0.95 for 10.75 ounces

Notice in Example 5.67 that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

Example 5.68

Translate the word phrase into an algebraic expression:

  1. 427 miles per h hours
  2. x students to 3 teachers
  3. y dollars for 18 hours

Try It 5.135

Translate the word phrase into an algebraic expression.

689 miles per h hours y parents to 22 students d dollars for 9 minutes

Try It 5.136

Translate the word phrase into an algebraic expression.

m miles per 9 hours x students to 8 buses y dollars for 40 hours

Section 5.6 Exercises

Practice Makes Perfect

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction.

403.

20 to 36

404.

20 to 32

405.

42 to 48

406.

45 to 54

407.

49 to 21

408.

56 to 16

409.

84 to 36

410.

6.4 to 0.8

411.

0.56 to 2.8

412.

1.26 to 4.2

413.

123 to 256

414.

134 to 258

415.

416 to 313

416.

535 to 335

417.

$18 to $63

418.

$16 to $72

419.

$1.21 to $0.44

420.

$1.38 to $0.69

421.

28 ounces to 84 ounces

422.

32 ounces to 128 ounces

423.

12 feet to 46 feet

424.

15 feet to 57 feet

425.

246 milligrams to 45 milligrams

426.

304 milligrams to 48 milligrams

427.

total cholesterol of 175 to HDL cholesterol of 45

428.

total cholesterol of 215 to HDL cholesterol of 55

429.

27 inches to 1 foot

430.

28 inches to 1 foot

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction.

431.

140 calories per 12 ounces

432.

180 calories per 16 ounces

433.

8.2 pounds per 3 square inches

434.

9.5 pounds per 4 square inches

435.

488 miles in 7 hours

436.

527 miles in 9 hours

437.

$595 for 40 hours

438.

$798 for 40 hours

Find Unit Rates

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

439.

140 calories per 12 ounces

440.

180 calories per 16 ounces

441.

8.2 pounds per 3 square inches

442.

9.5 pounds per 4 square inches

443.

488 miles in 7 hours

444.

527 miles in 9 hours

445.

$595 for 40 hours

446.

$798 for 40 hours

447.

576 miles on 18 gallons of gas

448.

435 miles on 15 gallons of gas

449.

43 pounds in 16 weeks

450.

57 pounds in 24 weeks

451.

46 beats in 0.5 minute

452.

54 beats in 0.5 minute

453.

The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour?

454.

The pressroom at a printing plant prints 540,000 sections in 12 hours. How many sections are printed per hour?

Find Unit Price

In the following exercises, find the unit price. Round to the nearest cent.

455.

Soap bars at 8 for $8.69

456.

Soap bars at 4 for $3.39

457.

Women’s sports socks at 6 pairs for $7.99

458.

Men’s dress socks at 3 pairs for $8.49

459.

Snack packs of cookies at 12 for $5.79

460.

Granola bars at 5 for $3.69

461.

CD-RW discs at 25 for $14.99

462.

CDs at 50 for $4.49

463.

The grocery store has a special on macaroni and cheese. The price is $3.87 for 3 boxes. How much does each box cost?

464.

The pet store has a special on cat food. The price is $4.32 for 12 cans. How much does each can cost?

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

465.

Mouthwash, 50.7-ounce size for $6.99 or 33.8-ounce size for $4.79

466.

Toothpaste, 6 ounce size for $3.19 or 7.8-ounce size for $5.19

467.

Breakfast cereal, 18 ounces for $3.99 or 14 ounces for $3.29

468.

Breakfast Cereal, 10.7 ounces for $2.69 or 14.8 ounces for $3.69

469.

Ketchup, 40-ounce regular bottle for $2.99 or 64-ounce squeeze bottle for $4.39

470.

Mayonnaise 15-ounce regular bottle for $3.49 or 22-ounce squeeze bottle for $4.99

471.

Cheese $6.49 for 1 lb. block or $3.39 for 12 lb. block

472.

Candy $10.99 for a 1 lb. bag or $2.89 for 14 lb. of loose candy

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

473.

793 miles per p hours

474.

78 feet per r seconds

475.

$3 for 0.5 lbs.

476.

j beats in 0.5 minutes

477.

105 calories in x ounces

478.

400 minutes for m dollars

479.

the ratio of y and 5x

480.

the ratio of 12x and y

Everyday Math

481.

One elementary school in Ohio has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate.

482.

The average American produces about 1,600 pounds of paper trash per year (365 days). How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)

483.

A popular fast food burger weighs 7.5 ounces and contains 540 calories, 29 grams of fat, 43 grams of carbohydrates, and 25 grams of protein. Find the unit rate of calories per ounce grams of fat per ounce grams of carbohydrates per ounce grams of protein per ounce. Round to two decimal places.

484.

A 16-ounce chocolate mocha coffee with whipped cream contains 470 calories, 18 grams of fat, 63 grams of carbohydrates, and 15 grams of protein. Find the unit rate of calories per ounce grams of fat per ounce grams of carbohydrates per ounce grams of protein per ounce.

Writing Exercises

485.

Would you prefer the ratio of your income to your friend’s income to be 3/1 or 1/3? Explain your reasoning.

486.

The parking lot at the airport charges $0.75 for every 15 minutes. How much does it cost to park for 1 hour? Explain how you got your answer to part . Was your reasoning based on the unit cost or did you use another method?

487.

Kathryn ate a 4-ounce cup of frozen yogurt and then went for a swim. The frozen yogurt had 115 calories. Swimming burns 422 calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.

488.

Mollie had a 16-ounce cappuccino at her neighborhood coffee shop. The cappuccino had 110 calories. If Mollie walks for one hour, she burns 246 calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

After reviewing this checklist, what will you do to become confident for all objectives?

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