Loading [MathJax]/jax/input/MathML/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5.7: Ratios and Rate

( \newcommand{\kernel}{\mathrm{null}\,}\)

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.

Answer

15 to 2715 to 27
Write as a fraction with the first number in the numerator and the second in the denominator. 15271527
Simplify the fraction. 5959
45 to 1845 to 18
Write as a fraction with the first number in the numerator and the second in the denominator. 45184518
Simplify. 5252

We leave the ratio in as an improper fraction.

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
Answer

4.8 to 11.24.8 to 11.2
Write as a fraction. 4.811.24.811.2
Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right. 4811248112
Simplify. 3737

So 4.8to11.24.8to11.2 is equivalent to 37.37.


The numerator has one decimal place and the denominator has 2.2. To clear both decimals we need to move the decimal 22 places to the right.
2.7to0.542.7to0.54
Write as a fraction. 2.70.542.70.54
Move both decimals right two places. 2705427054
Simplify. 5151

So 2.7to0.542.7to0.54 is equivalent to 51.51.

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.

Answer

114to238114to238
Write as a fraction. 114238114238
Convert the numerator and denominator to improper fractions. 5419854198
Rewrite as a division of fractions. 54÷19854÷198
Invert the divisor and multiply. 54·81954·819
Simplify. 10191019

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?

Answer

First, write the words that express the ratio. We want to know the ratio of Hector's total cholesterol to his HDL cholesterol.

Write as a fraction. total cholesterolHDL cholesteroltotal cholesterolHDL cholesterol
Substitute the values. 2493924939
Simplify. 83138313

Is Hector's cholesterol ratio ok? If we divide 8383 by 1313 we obtain approximately 6.4,6.4, so 83136.41.83136.41. Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.

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?

Answer

In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.

Write the words that express the ratio.

Ratio of the rise to the run
Write the ratio as a fraction. riserunriserun
Substitute in the given values. 1 inch1 foot1 inch1 foot
Convert 1 foot to inches. 1 inch12 inches1 inch12 inches
Simplify, dividing out common factors and units. 112112

So the ratio of rise to run is 11 to 12.12. This means that the ramp should rise 11 inch for every 1212 inches of horizontal run to comply with the guidelines.

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.

Answer

525 miles in 9 hours525 miles in 9 hours
Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator. 525 miles9 hours525 miles9 hours
175 miles3 hours175 miles3 hours

So 525525 miles in 99 hours is equivalent to 175 miles3 hours.175 miles3 hours.

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?

Answer

Start with a rate of dollars to hours. Then divide. $384 last week for 32 hours$384 last week for 32 hours
Write as a rate. $38432 hours$38432 hours
Divide the numerator by the denominator. $121 hour$121 hour
Rewrite as a rate. $12/hour$12/hour

Anita’s hourly pay rate is $12$12 per hour.

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?

Answer

Start with a rate of miles to gallons. Then divide.

455 miles to 14 gallons of gas455 miles to 14 gallons of gas
Write as a rate. 455 miles14 gallons455 miles14 gallons
Divide 455 by 14 to get the unit rate. 32.5 miles1 gallon32.5 miles1 gallon

Sven’s car gets 32.532.5 miles/gallon, or 32.532.5 mpg.

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?

Answer

What are we asked to find? We are asked to find the unit price, which is the price per bottle.

Write as a rate. $3.9924 bottles$3.9924 bottles
Divide to find the unit price. $0.166251 bottle$0.166251 bottle
Round the result to the nearest penny. $0.171 bottle$0.171 bottle

The unit price is approximately $0.17$0.17 per bottle. Each bottle costs about $0.17.$0.17.

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 detergent has the lowest cost per load?

Answer

To compare the prices, we first find the unit price for each type of detergent.

  Liquid Powder
Write as a rate. $14.9964 loads$14.9964 loads $15.9980 loads$15.9980 loads
Find the unit price. $0.234…1 load$0.234…1 load $0.199…1 load$0.199…1 load
Round to the nearest cent. $0.23/load(23 cents per load.)$0.23/load(23 cents per load.) $0.20/load(20 cents per load)$0.20/load(20 cents per load)

Now we compare the unit prices. The unit price of the liquid detergent is about $0.23$0.23 per load and the unit price of the powder detergent is about $0.20$0.20 per load. The powder is the better buy.

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
Answer

427 miles perhhours427 miles perhhours
Write as a rate. 427 mileshhours427 mileshhours
xstudents to 3 teachersxstudents to 3 teachers
Write as a rate. xstudents3 teachersxstudents3 teachers
ydollars for 18 hoursydollars for 18 hours
Write as a rate. $y18 hours$y18 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

Media

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?


This page titled 5.7: Ratios and Rate is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

  • Was this article helpful?

Support Center

How can we help?