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Prealgebra 2e

5.6 Ratios and Rate

Prealgebra 2e5.6 Ratios and Rate
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

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: 1624.1624.
If you missed this problem, review Example 4.19.

Be Prepared 5.17

Divide: 2.76÷11.5.2.76÷11.5.
If you missed this problem, review Example 5.19.

Be Prepared 5.18

Simplify: 112234.112234.
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 aa and bb, the ratio is written as atob,ab,ora:b.atob,ab,ora:b.

Ratios

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of aa to bb is written atob,ab,ora:b.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 4141 instead of simplifying it to 44 so that we can see the two parts of the ratio.

Example 5.58

Write each ratio as a fraction: 15to2715to2745to18.45to18.

Try It 5.115

Write each ratio as a fraction: 21to5621to5648to32.48to32.

Try It 5.116

Write each ratio as a fraction: 27to7227to7251to34.51to34.

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 0.8to0.05.0.8to0.05. We can write it as a fraction with decimals and then multiply the numerator and denominator by 100100 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=8100.8=810 and 0.05=5100.0.05=5100. The least common denominator of 810810 and 51005100 is 100.100. By multiplying the numerator and denominator of 0.80.050.80.05 by 100,100, 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. 805805
Simplify. 161161

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. 4.8to11.24.8to11.2
  2. 2.7to0.542.7to0.54

Try It 5.117

Write each ratio as a fraction: 4.6to11.54.6to11.52.3to0.69.2.3to0.69.

Try It 5.118

Write each ratio as a fraction: 3.4to15.33.4to15.33.4to0.68.3.4to0.68.

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 114to238114to238 as a fraction.

Try It 5.119

Write each ratio as a fraction: 134to258.134to258.

Try It 5.120

Write each ratio as a fraction: 118to234.118to234.

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 55 to 11 is considered good.

Example 5.61

Hector's total cholesterol is 249249 mg/dl and his HDL cholesterol is 3939 mg/dl. Find the ratio of his total cholesterol to his HDL cholesterol. Assuming that a ratio less than 55 to 11 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 185185 mg/dL and HDL cholesterol is 4040 mg/dL.

Try It 5.122

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

Total cholesterol is 204204 mg/dL and HDL cholesterol is 3838 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 11 inch for every 11 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: 3232 inches to 11 foot.

Try It 5.124

Find the ratio of the first length to the second length: 11 foot to 5454 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 120120 miles in 22 hours, 160160 words in 44 minutes, and $5$5 dollars per 6464 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 525525 miles in 99 hours. Write this rate as a fraction.

Try It 5.125

Write the rate as a fraction: 492492 miles in 88 hours.

Try It 5.126

Write the rate as a fraction: 242242 miles in 66 hours.

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of 175 miles3 hours.175 miles3 hours. This tells us that every three hours, Bob will travel 175175 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 11 unit is referred to as a unit rate.

Unit Rate

A unit rate is a rate with denominator of 11 unit.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of 6868 miles per hour we mean that we travel 6868 miles in 11 hour. We would write this rate as 6868 miles/hour (read 6868 miles per hour). The common abbreviation for this is 6868 mph. Note that when no number is written before a unit, it is assumed to be 1.1.

So 6868 miles/hour really means 68 miles/1 hour.68 miles/1 hour.

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

Example Rate Write Abbreviate Read
6868 miles in 11 hour 68 miles1 hour68 miles1 hour 6868 miles/hour 6868 mph 68 miles per hour68 miles per hour
3636 miles to 11 gallon 36 miles1 gallon36 miles1 gallon 3636 miles/gallon 3636 mpg 36 miles per gallon36 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$12.50 for each hour you work, you could write that your hourly (unit) pay rate is $12.50/hour$12.50/hour (read $12.50$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 1.1.

Example 5.64

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

Try It 5.127

Find the unit rate: $630$630 for 3535 hours.

Try It 5.128

Find the unit rate: $684$684 for 3636 hours.

Example 5.65

Sven drives his car 455455 miles, using 1414 gallons of gasoline. How many miles per gallon does his car get?

Try It 5.129

Find the unit rate: 423423 miles to 1818 gallons of gas.

Try It 5.130

Find the unit rate: 406406 miles to 14.514.5 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$3.99 for a case of 2424 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-pack24-pack of juice boxes for $6.99$6.99

Try It 5.132

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

24-pack24-pack of bottles of ice tea for $12.72$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$14.99 for 6464 loads of laundry and the same brand of powder detergent is priced at $15.99$15.99 for 8080 loads.

Which is the better buy, the liquid or the powder detergent?

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$4.59 for 4040 count, or Brand B Storage Bags, $3.99$3.99 for 3030 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$1.89 for 2626 ounces, or Brand D Chicken Noodle Soup, $0.95$0.95 for 10.7510.75 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. 427427 miles per hh hours
  2. xx students to 33 teachers
  3. yy dollars for 1818 hours
Try It 5.135

Translate the word phrase into an algebraic expression.

689689 miles per hh hours yy parents to 2222 students dd dollars for 99 minutes

Try It 5.136

Translate the word phrase into an algebraic expression.

mm miles per 99 hours xx students to 88 buses yy dollars for 4040 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.

2020 to 3636

404.

2020 to 3232

405.

4242 to 4848

406.

4545 to 5454

407.

4949 to 2121

408.

5656 to 1616

409.

8484 to 3636

410.

6.46.4 to 0.80.8

411.

0.560.56 to 2.82.8

412.

1.261.26 to 4.24.2

413.

123123 to 256256

414.

134134 to 258258

415.

416416 to 313313

416.

535535 to 335335

417.

$18$18 to $63$63

418.

$16$16 to $72$72

419.

$1.21$1.21 to $0.44$0.44

420.

$1.38$1.38 to $0.69$0.69

421.

2828 ounces to 8484 ounces

422.

3232 ounces to 128128 ounces

423.

1212 feet to 4646 feet

424.

1515 feet to 5757 feet

425.

246246 milligrams to 4545 milligrams

426.

304304 milligrams to 4848 milligrams

427.

total cholesterol of 175175 to HDL cholesterol of 4545

428.

total cholesterol of 215215 to HDL cholesterol of 5555

429.

2727 inches to 11 foot

430.

2828 inches to 11 foot

Write a Rate as a Fraction

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

431.

140140 calories per 1212 ounces

432.

180180 calories per 1616 ounces

433.

8.28.2 pounds per 33 square inches

434.

9.59.5 pounds per 44 square inches

435.

488488 miles in 77 hours

436.

527527 miles in 99 hours

437.

$595$595 for 4040 hours

438.

$798$798 for 4040 hours

Find Unit Rates

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

439.

140140 calories per 1212 ounces

440.

180180 calories per 1616 ounces

441.

8.28.2 pounds per 33 square inches

442.

9.59.5 pounds per 44 square inches

443.

488488 miles in 77 hours

444.

527527 miles in 99 hours

445.

$595$595 for 4040 hours

446.

$798$798 for 4040 hours

447.

576576 miles on 1818 gallons of gas

448.

435435 miles on 1515 gallons of gas

449.

4343 pounds in 1616 weeks

450.

5757 pounds in 2424 weeks

451.

4646 beats in 0.50.5 minute

452.

5454 beats in 0.50.5 minute

453.

The bindery at a printing plant assembles 96,00096,000 magazines in 1212 hours. How many magazines are assembled in one hour?

454.

The pressroom at a printing plant prints 540,000540,000 sections in 1212 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 88 for $8.69$8.69

456.

Soap bars at 44 for $3.39$3.39

457.

Women’s sports socks at 66 pairs for $7.99$7.99

458.

Men’s dress socks at 33 pairs for $8.49$8.49

459.

Snack packs of cookies at 1212 for $5.79$5.79

460.

Granola bars at 55 for $3.69$3.69

461.

CD-RW discs at 2525 for $14.99$14.99

462.

CDs at 5050 for $4.49$4.49

463.

The grocery store has a special on macaroni and cheese. The price is $3.87$3.87 for 33 boxes. How much does each box cost?

464.

The pet store has a special on cat food. The price is $4.32$4.32 for 1212 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-ounce50.7-ounce size for $6.99$6.99 or 33.8-ounce33.8-ounce size for $4.79$4.79

466.

Toothpaste, 66 ounce size for $3.19$3.19 or 7.8-ounce7.8-ounce size for $5.19$5.19

467.

Breakfast cereal, 1818 ounces for $3.99$3.99 or 1414 ounces for $3.29$3.29

468.

Breakfast Cereal, 10.710.7 ounces for $2.69$2.69 or 14.814.8 ounces for $3.69$3.69

469.

Ketchup, 40-ounce40-ounce regular bottle for $2.99$2.99 or 64-ounce64-ounce squeeze bottle for $4.39$4.39

470.

Mayonnaise 15-ounce15-ounce regular bottle for $3.49$3.49 or 22-ounce22-ounce squeeze bottle for $4.99$4.99

471.

Cheese $6.49$6.49 for 11 lb. block or $3.39$3.39 for 1212 lb. block

472.

Candy $10.99$10.99 for a 11 lb. bag or $2.89$2.89 for 1414 lb. of loose candy

Translate Phrases to Expressions with Fractions

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

473.

793793 miles per pp hours

474.

7878 feet per rr seconds

475.

$3$3 for 0.50.5 lbs.

476.

jj beats in 0.50.5 minutes

477.

105105 calories in xx ounces

478.

400400 minutes for mm dollars

479.

the ratio of yy and 5x5x

480.

the ratio of 12x12x and yy

Everyday Math

481.

One elementary school in Ohio has 684684 students and 4545 teachers. Write the student-to-teacher ratio as a unit rate.

482.

The average American produces about 1,6001,600 pounds of paper trash per year (365 days).(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 7.57.5 ounces and contains 540540 calories, 2929 grams of fat, 4343 grams of carbohydrates, and 2525 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-ounce16-ounce chocolate mocha coffee with whipped cream contains 470470 calories, 1818 grams of fat, 6363 grams of carbohydrates, and 1515 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 3/13/1 or 1/3?1/3? Explain your reasoning.

486.

The parking lot at the airport charges $0.75$0.75 for every 1515 minutes. How much does it cost to park for 11 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-ounce4-ounce cup of frozen yogurt and then went for a swim. The frozen yogurt had 115115 calories. Swimming burns 422422 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-ounce16-ounce cappuccino at her neighborhood coffee shop. The cappuccino had 110110 calories. If Mollie walks for one hour, she burns 246246 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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