Ratios and Rate

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 $a to b, \frac{a}{b}, or a : b .$

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.

Write each ratio as a fraction: ⓐ $15 to 27$ ⓑ $45 to 18 .$

Solution

This table illustrates the step-by-step process of converting a given ratio into a simplified fraction.
ⓐ
	$15 to 27$
Write as a fraction with the first number in the numerator and the second in the denominator.	$\frac{15}{27}$
Simplify the fraction.	$\frac{5}{9}$

This table demonstrates converting and simplifying the ratio 45 to 18 into a fraction, showing the steps involved.
ⓑ
	$45 to 18$
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.

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:

Illustrates dividing decimals by converting to whole numbers and simplifying the resulting fraction.

"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.

Write each ratio as a fraction of whole numbers:

ⓐ $4.8 to 11.2$
ⓑ $2.7 to 0.54$

Solution

This table illustrates the step-by-step process of converting the decimal ratio 4.8 to 11.2 into its simplest fractional form, resulting in 3/7.
ⓐ $4.8 to 11.2$
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 $\frac{3}{7} .$

Steps to simplify a fraction with decimals, demonstrating how to clear decimals and reduce the fraction to its simplest form.
ⓑ 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 $\frac{5}{1} .$

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

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

Solution

This table illustrates the step-by-step process of simplifying a ratio of two mixed numbers into a single simplified fraction.
	$1 \frac{1}{4} to 2 \frac{3}{8}$
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.	$\frac{10}{19}$

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.

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.

Step-by-step process for expressing and simplifying a ratio of cholesterol values as a fraction.
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.

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.

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.

Step-by-step calculation of the rise to run ratio, demonstrating unit conversion and simplification.
	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.

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.

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.

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

Solution

Demonstrates converting a rate into a fraction and simplifying it.
	$525 miles in 9 hours$
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 $\frac{175 miles}{3 hours} .$

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 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 $1 .$

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

Solution

Steps demonstrating how to calculate an hourly rate from a given total earning and hours worked.
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.

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.

This table illustrates the steps to convert a given ratio of miles to gallons into its equivalent unit rate.
	$455 miles to 14 gallons of gas$
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.

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.

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.

Steps for calculating unit price per bottle, from initial rate to rounded final amount.
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 .$

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.

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?

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.

Notice in Example 10 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.

Translate the word phrase into an algebraic expression:

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

Solution

This table demonstrates how to convert a verbal rate description into its mathematical fractional representation.
ⓐ
	$427 miles per h hours$
Write as a rate.	$\frac{427 miles}{h hours}$

This table demonstrates how to convert a verbal ratio, such as 'x students to 3 teachers', into a fractional rate expression.
ⓑ
	$x students to 3 teachers$
Write as a rate.	$\frac{x students}{3 teachers}$

Demonstrates converting 'y dollars for 18 hours' into a mathematical rate.
ⓒ
	$y dollars for 18 hours$
Write as a rate.	$\frac{$ y}{18 hours}$

Practice Makes Perfect

Write a Ratio as a Fraction

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

$20$ to $36$

Solution

$\frac{5}{9}$

$20$ to $32$

$42$ to $48$

Solution

$\frac{7}{8}$

$45$ to $54$

$49$ to $21$

Solution

$\frac{7}{3}$

$56$ to $16$

$84$ to $36$

Solution

$\frac{7}{3}$

$6.4$ to $0.8$

$0.56$ to $2.8$

Solution

$\frac{1}{5}$

$1.26$ to $4.2$

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

Solution

$\frac{10}{17}$

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

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

Solution

$\frac{5}{4}$

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

$$18$ to $$63$

Solution

$\frac{2}{7}$

$$16$ to $$72$

$$1.21$ to $$0.44$

Solution

$\frac{11}{4}$

$$1.38$ to $$0.69$

$28$ ounces to $84$ ounces

Solution

$\frac{1}{3}$

$32$ ounces to $128$ ounces

$12$ feet to $46$ feet

Solution

$\frac{6}{23}$

$15$ feet to $57$ feet

$246$ milligrams to $45$ milligrams

Solution

$\frac{82}{15}$

$304$ milligrams to $48$ milligrams

total cholesterol of $175$ to HDL cholesterol of $45$

Solution

$\frac{35}{9}$

total cholesterol of $215$ to HDL cholesterol of $55$

$27$ inches to $1$ foot

Solution

$\frac{9}{4}$

$28$ inches to $1$ foot

Write a Rate as a Fraction

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

$140$ calories per $12$ ounces

Solution

$\frac{35 calories}{3 ounces}$

$180$ calories per $16$ ounces

$8.2$ pounds per $3$ square inches

Solution

$\frac{41 lbs}{15 sq. in .}$

$9.5$ pounds per $4$ square inches

$488$ miles in $7$ hours

Solution

$\frac{488 miles}{7 hours}$

$527$ miles in $9$ hours

$$595$ for $40$ hours

Solution

$\frac{$119}{8 hours}$

$$798$ for $40$ hours

Find Unit Rates

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

$140$ calories per $12$ ounces

Solution

11.67 calories/ounce

$180$ calories per $16$ ounces

$8.2$ pounds per $3$ square inches

Solution

2.73 lbs./sq. in.

$9.5$ pounds per $4$ square inches

$488$ miles in $7$ hours

Solution

69.71 mph

$527$ miles in $9$ hours

$$595$ for $40$ hours

Solution

$14.88/hour

$$798$ for $40$ hours

$576$ miles on $18$ gallons of gas

Solution

32 mpg

$435$ miles on $15$ gallons of gas

$43$ pounds in $16$ weeks

Solution

2.69 lbs./week

$57$ pounds in $24$ weeks

$46$ beats in $0.5$ minute

Solution

92 beats/minute

$54$ beats in $0.5$ minute

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

Solution

8,000

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.

Soap bars at $8$ for $$8.69$

Solution

$1.09/bar

Soap bars at $4$ for $$3.39$

Women’s sports socks at $6$ pairs for $$7.99$

Solution

$1.33/pair

Men’s dress socks at $3$ pairs for $$8.49$

Snack packs of cookies at $12$ for $$5.79$

Solution

$0.48/pack

Granola bars at $5$ for $$3.69$

CD-RW discs at $25$ for $$14.99$

Solution

$0.60/disc

CDs at $50$ for $$4.49$

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

Solution

$1.29/box

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.

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

Solution

The 50.7-ounce size costs $0.138 per ounce. The 33.8-ounce size costs $0.142 per ounce. The 50.7-ounce size is the better buy.

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

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

Solution

The 18-ounce size costs $0.222 per ounce. The 14-ounce size costs $0.235 per ounce. The 18-ounce size is a better buy.

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

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

Solution

The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy.

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

Cheese $$6.49$ for $1$ lb. block or $$3.39$ for $\frac{1}{2}$ lb. block

Solution

The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy.

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.

$793$ miles per $p$ hours

Solution

$\frac{793 miles}{p hours}$

$78$ feet per $r$ seconds

$$3$ for $0.5$ lbs.

Solution

$\frac{$3}{0.5 lbs .}$

$j$ beats in $0.5$ minutes

$105$ calories in $x$ ounces

Solution

$\frac{105 calories}{x ounces}$

$400$ minutes for $m$ dollars

the ratio of $y$ and $5 x$

Solution

$\frac{y}{5 x}$

the ratio of $12 x$ and $y$

Everyday Math

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

Solution

15.2 students per teacher

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.)

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.

Solution

ⓐ 72 calories/ounce
ⓑ 3.87 grams of fat/ounce
ⓓ 3.33 grams protein/ounce

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

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

Solution

Answers will vary.

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?

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.

Solution

Kathryn should swim for approximately 16.35 minutes. Explanations will vary.

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?

ratio

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of

a

b

is written

a

b

\frac{a}{b}

, or

a : b

rate

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

unit rate

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

unit price

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