5.6 Ratios and Rate - Prealgebra 2e

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: $\frac{16}{24} .$
If you missed this problem, review Example 4.19.

Be Prepared 5.17

Divide: $2.76 \div 11.5 .$
If you missed this problem, review Example 5.19.

Be Prepared 5.18

Simplify: $\frac{1 \frac{1}{2}}{2 \frac{3}{4}} .$
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 $a to b, \frac{a}{b}, or a : b .$

Ratios

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a to b, \frac{a}{b}, or a : b .$

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 $\frac{4}{1}$ 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: ⓐ $15 to 27$ ⓑ $45 to 18 .$

Solution

ⓐ
	$15 to 27$
Write as a fraction with the first number in the numerator and the second in the denominator.	$\frac{15}{27}$
Simplify the fraction.	$\frac{5}{9}$

ⓑ
	$45 to 18$
Write as a fraction with the first number in the numerator and the second in the denominator.	$\frac{45}{18}$
Simplify.	$\frac{5}{2}$

We leave the ratio in ⓑ as an improper fraction.

Try It 5.115

Write each ratio as a fraction: ⓐ $21 to 56$ ⓑ $48 to 32 .$

Try It 5.116

Write each ratio as a fraction: ⓐ $27 to 72$ ⓑ $51 to 34 .$

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 $0.8 to 0.05 .$ 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 $0.8 = \frac{8}{10}$ and $0.05 = \frac{5}{100} .$ The least common denominator of $\frac{8}{10}$ and $\frac{5}{100}$ is $100 .$ By multiplying the numerator and denominator of $\frac{0.8}{0.05}$ 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:


"Move" the decimal 2 places.	$\frac{80}{5}$
Simplify.	$\frac{16}{1}$

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:

ⓐ $4.8 to 11.2$
ⓑ $2.7 to 0.54$

Solution

ⓐ $4.8 to 11.2$
Write as a fraction.	$\frac{4.8}{11.2}$
Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right.	$\frac{48}{112}$
Simplify.	$\frac{3}{7}$

So $4.8 to 11.2$ is equivalent to $\frac{3}{7} .$

ⓑ The numerator has one decimal place and the denominator has $2 .$ To clear both decimals we need to move the decimal $2$ places to the right. $2.7 to 0.54$
Write as a fraction.	$\frac{2.7}{0.54}$
Move both decimals right two places.	$\frac{270}{54}$
Simplify.	$\frac{5}{1}$

So $2.7 to 0.54$ is equivalent to $\frac{5}{1} .$

Try It 5.117

Write each ratio as a fraction: ⓐ $4.6 to 11.5$ ⓑ $2.3 to 0.69 .$

Try It 5.118

Write each ratio as a fraction: ⓐ $3.4 to 15.3$ ⓑ $3.4 to 0.68 .$

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 $1 \frac{1}{4} to 2 \frac{3}{8}$ as a fraction.

Solution

	$1 \frac{1}{4} to 2 \frac{3}{8}$
Write as a fraction.	$\frac{1 \frac{1}{4}}{2 \frac{3}{8}}$
Convert the numerator and denominator to improper fractions.	$\frac{\frac{5}{4}}{\frac{19}{8}}$
Rewrite as a division of fractions.	$\frac{5}{4} \div \frac{19}{8}$
Invert the divisor and multiply.	$\frac{5}{4} \cdot \frac{8}{19}$
Simplify.	$\frac{10}{19}$

Try It 5.119

Write each ratio as a fraction: $1 \frac{3}{4} to 2 \frac{5}{8} .$

Try It 5.120

Write each ratio as a fraction: $1 \frac{1}{8} to 2 \frac{3}{4} .$

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?

Solution

ⓐ 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.	$\frac{total cholesterol}{HDL cholesterol}$
Substitute the values.	$\frac{249}{39}$
Simplify.	$\frac{83}{13}$

ⓑ Is Hector's cholesterol ratio ok? If we divide $83$ by $13$ we obtain approximately $6.4,$ so $\frac{83}{13} \approx \frac{6.4}{1} .$ 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 $185$ 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?

Solution

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.	$\frac{rise}{run}$
Substitute in the given values.	$\frac{1 inch}{1 foot}$
Convert 1 foot to inches.	$\frac{1 inch}{12 inches}$
Simplify, dividing out common factors and units.	$\frac{1}{12}$

So the ratio of rise to run is $1$ to $12 .$ This means that the ramp should rise $1$ inch for every $12$ inches of horizontal run to comply with the guidelines.

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.

Solution

	$525 miles in 9 hours$
Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator.	$\frac{525 miles}{9 hours}$
	$\frac{175 miles}{3 hours}$

So $525$ miles in $9$ hours is equivalent to $\frac{175 miles}{3 hours} .$

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 $\frac{175 miles}{3 hours} .$ 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	$\frac{68 miles}{1 hour}$	$68$ miles/hour	$68$ mph	$68 miles per hour$
$36$ miles to $1$ gallon	$\frac{36 miles}{1 gallon}$	$36$ 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?

Solution

Start with a rate of dollars to hours. Then divide.	$$384 last week for 32 hours$
Write as a rate.	$\frac{$384}{32 hours}$
Divide the numerator by the denominator.	$\frac{$12}{1 hour}$
Rewrite as a rate.	$$12 / hour$

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

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?

Solution

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

	$455 miles to 14 gallons of gas$
Write as a rate.	$\frac{455 miles}{14 gallons}$
Divide 455 by 14 to get the unit rate.	$\frac{32.5 miles}{1 gallon}$

Sven’s car gets $32.5$ miles/gallon, or $32.5$ mpg.

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?

Solution

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

Write as a rate.	$\frac{$3.99}{24 bottles}$
Divide to find the unit price.	$\frac{$0.16625}{1 bottle}$
Round the result to the nearest penny.	$\frac{$0.17}{1 bottle}$

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

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

Solution

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

	Liquid	Powder
Write as a rate.	$\frac{$14.99}{64 loads}$	$\frac{$15.99}{80 loads}$
Find the unit price.	$\frac{$0.234…}{1 load}$	$\frac{$0.199…}{1 load}$
Round to the nearest cent.	$\begin{matrix} $0.23/load \\ (23 cents per load.) \end{matrix}$	$\begin{matrix} $0.20/load \\ (20 cents per load) \end{matrix}$

Now we compare the unit prices. The unit price of the liquid detergent is about $$0.23$ per load and the unit price of the powder detergent is about $$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, $$4.59$ for $40$ count, or Brand B Storage Bags, $$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:

ⓐ $427$ miles per $h$ hours
ⓑ $x$ students to $3$ teachers
ⓒ $y$ dollars for $18$ hours

Solution

ⓐ
	$427 miles per h hours$
Write as a rate.	$\frac{427 miles}{h hours}$

ⓑ
	$x students to 3 teachers$
Write as a rate.	$\frac{x students}{3 teachers}$

ⓒ
	$y dollars for 18 hours$
Write as a rate.	$\frac{$ y}{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

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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.

$1 \frac{2}{3}$ to $2 \frac{5}{6}$

414.

$1 \frac{3}{4}$ to $2 \frac{5}{8}$

415.

$4 \frac{1}{6}$ to $3 \frac{1}{3}$

416.

$5 \frac{3}{5}$ to $3 \frac{3}{5}$

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 $\frac{1}{2}$ lb. block

472.

Candy $$10.99$ for a $1$ lb. bag or $$2.89$ for $\frac{1}{4}$ 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 $5 x$

480.

the ratio of $12 x$ 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?