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Ratios and Rate

Write a ratio as a fraction

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

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

Try it.

Write each ratio as a fraction: ⓐ $\ 15\ \text{to}\ 27\$ⓑ $\ 45\ \text{to}\ 18.$

Solution

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

ⓑ
	$\text{45 to 18}$
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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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 $\text{\$5}$ 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.

Example

Try it.

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

Solution

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

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

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of $\frac{\text{175 miles}}{\text{3 hours}}.$ 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 $\text{68 miles/1 hour.}$

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	$\frac{\text{68 miles}}{\text{1 hour}}$	$68$ miles/hour	$68$ mph	$\text{68 miles per hour}$
$36$ miles to $1$ gallon	$\frac{\text{36 miles}}{\text{1 gallon}}$	$36$ miles/gallon	$36$ mpg	$\text{36 miles per gallon}$

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 $\text{\$12.50}$ for each hour you work, you could write that your hourly (unit) pay rate is $\text{\$12.50/hour}$ (read $\text{\$12.50}$ 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

Try it.

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

Solution

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

Anita’s hourly pay rate is $\text{\$12}$ per hour.

Example

Try it.

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.

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

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

Condensed — the full section is in OpenStax Prealgebra 2e.

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.

Example

Try it.

The grocery store charges $\text{\$3.99}$ 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.	$\frac{\$3.99}{\text{24 bottles}}$
Divide to find the unit price.	$\frac{\$0.16625}{\text{1 bottle}}$
Round the result to the nearest penny.	$\frac{\$0.17}{\text{1 bottle}}$

The unit price is approximately $\text{\$0.17}$ per bottle. Each bottle costs about $\text{\$0.17}.$

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.

Example

Try it.

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $\text{\$14.99}$ for $64$ loads of laundry and the same brand of powder detergent is priced at $\text{\$15.99}$ 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.	$\frac{\text{\$14.99}}{\text{64 loads}}$	$\frac{\text{\$15.99}}{\text{80 loads}}$
Find the unit price.	$\frac{\text{\$0.234\ldots }}{\text{1 load}}$	$\frac{\text{\$0.199\ldots }}{\text{1 load}}$
Round to the nearest cent.	$\begin{array}{l}\text{\$0.23/load} \\ \text{(23 cents per load.)}\end{array}$	$\begin{array}{l}\text{\$0.20/load} \\ \text{(20 cents per load)}\end{array}$

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

Notice in 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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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

Try it.

Translate the word phrase into an algebraic expression:

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

Solution

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

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

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

Ratios and Rate

Write a Ratio as a Fraction

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

Try it.

$20$ to $36$

Solution

$\frac{5}{9}$

Try it.

$20$ to $32$

Try it.

$42$ to $48$

Solution

$\frac{7}{8}$

Try it.

$45$ to $54$

Try it.

$49$ to $21$

Solution

$\frac{7}{3}$

Try it.

$56$ to $16$

Try it.

$84$ to $36$

Solution

$\frac{7}{3}$

Try it.

$6.4$ to $0.8$

Try it.

$0.56$ to $2.8$

Solution

$\frac{1}{5}$

Try it.

$1.26$ to $4.2$

Try it.

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

Solution

$\frac{10}{17}$

Try it.

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

Try it.

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

Solution

$\frac{5}{4}$

Try it.

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

Try it.

$\text{\$18}$ to $\text{\$63}$

Solution

$\frac{2}{7}$

Try it.

$\text{\$16}$ to $\text{\$72}$

Try it.

$\text{\$1.21}$ to $\text{\$0.44}$

Solution

$\frac{11}{4}$

Try it.

$\text{\$1.38}$ to $\text{\$0.69}$

Try it.

$28$ ounces to $84$ ounces

Solution

$\frac{1}{3}$

Try it.

$32$ ounces to $128$ ounces

Try it.

$12$ feet to $46$ feet

Solution

$\frac{6}{23}$

Try it.

$15$ feet to $57$ feet

Try it.

$246$ milligrams to $45$ milligrams

Solution

$\frac{82}{15}$

Try it.

$304$ milligrams to $48$ milligrams

Try it.

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

Solution

$\frac{35}{9}$

Try it.

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

Try it.

$27$ inches to $1$ foot

Solution

$\frac{9}{4}$

Try it.

$28$ inches to $1$ foot

Write a Rate as a Fraction

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

Try it.

$140$ calories per $12$ ounces

Solution

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

Try it.

$180$ calories per $16$ ounces

Try it.

$8.2$ pounds per $3$ square inches

Solution

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

Try it.

$9.5$ pounds per $4$ square inches

Try it.

$488$ miles in $7$ hours

Solution

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

Try it.

$527$ miles in $9$ hours

Try it.

$\text{\$595}$ for $40$ hours

Solution

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

Try it.

$\text{\$798}$ for $40$ hours

Find Unit Rates

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

Try it.

$140$ calories per $12$ ounces

Solution

11.67 calories/ounce

Try it.

$180$ calories per $16$ ounces

Try it.

$8.2$ pounds per $3$ square inches

Solution

2.73 lbs./sq. in.

Try it.

$9.5$ pounds per $4$ square inches

Try it.

$488$ miles in $7$ hours

Solution

69.71 mph

Try it.

$527$ miles in $9$ hours

Try it.

$\text{\$595}$ for $40$ hours

Solution

$14.88/hour

Try it.

$\text{\$798}$ for $40$ hours

Try it.

$576$ miles on $18$ gallons of gas

Solution

32 mpg

Try it.

$435$ miles on $15$ gallons of gas

Try it.

$43$ pounds in $16$ weeks

Solution

2.69 lbs./week

Try it.

$57$ pounds in $24$ weeks

Try it.

$46$ beats in $0.5$ minute

Solution

92 beats/minute

Try it.

$54$ beats in $0.5$ minute

Try it.

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

Solution

8,000

Try it.

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.

Try it.

Soap bars at $8$ for $\text{\$8.69}$

Solution

$1.09/bar

Try it.

Soap bars at $4$ for $\text{\$3.39}$

Try it.

Women’s sports socks at $6$ pairs for $\text{\$7.99}$

Solution

$1.33/pair

Try it.

Men’s dress socks at $3$ pairs for $\text{\$8.49}$

Try it.

Snack packs of cookies at $12$ for $\text{\$5.79}$

Solution

$0.48/pack

Try it.

Granola bars at $5$ for $\text{\$3.69}$

Try it.

CD-RW discs at $25$ for $\text{\$14.99}$

Solution

$0.60/disc

Try it.

CDs at $50$ for $\text{\$4.49}$

Try it.

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

Solution

$1.29/box

Try it.

The pet store has a special on cat food. The price is $\text{\$4.32}$ 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.

Try it.

Mouthwash, $\text{50.7-ounce}$ size for $\text{\$6.99}$ or $\text{33.8-ounce}$ size for $\text{\$4.79}$

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.

Try it.

Toothpaste, $6$ ounce size for $\text{\$3.19}$ or $7.8-ounce$ size for $\text{\$5.19}$

Try it.

Breakfast cereal, $18$ ounces for $\text{\$3.99}$ or $14$ ounces for $\text{\$3.29}$

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.

Try it.

Breakfast Cereal, $10.7$ ounces for $\text{\$2.69}$ or $14.8$ ounces for $\text{\$3.69}$

Try it.

Ketchup, $\text{40-ounce}$ regular bottle for $\text{\$2.99}$ or $\text{64-ounce}$ squeeze bottle for $\text{\$4.39}$

Solution

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

Try it.

Mayonnaise $\text{15-ounce}$ regular bottle for $\text{\$3.49}$ or $\text{22-ounce}$ squeeze bottle for $\text{\$4.99}$

Try it.

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

Solution

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

Try it.

Candy $\text{\$10.99}$ for a $1$ lb. bag or $\text{\$2.89}$ 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.

Try it.

$793$ miles per $p$ hours

Solution

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

Try it.

$78$ feet per $r$ seconds

Try it.

$\text{\$3}$ for $0.5$ lbs.

Solution

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

Try it.

$j$ beats in $0.5$ minutes

Try it.

$105$ calories in $x$ ounces

Solution

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

Try it.

$400$ minutes for $m$ dollars

Try it.

the ratio of $y$ and $5x$

Solution

$\frac{y}{5x}$

Try it.

the ratio of $12x$ and $y$

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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

Onthul het antwoord

$\frac{2}{3}$
Divide: $2.76\div 11.5.$
If you missed this problem, review .

Onthul het antwoord

$0.24$
Simplify: $\frac{1\frac{1}{2}}{2\frac{3}{4}}.$
If you missed this problem, review .

Onthul het antwoord

$\frac{6}{11}$
Write each ratio as a fraction: ⓐ $\ 15\ \text{to}\ 27\$ⓑ $\ 45\ \text{to}\ 18.$

Onthul het antwoord

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

ⓑ
$\text{45 to 18}$
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.
Write each ratio as a fraction: ⓐ $\ 21\ \text{to}\ 56\$ⓑ $\ 48\ \text{to}\ 32.$
Onthul het antwoord
1. ⓐ $\ \frac{3}{8}$
2. ⓑ $\ \frac{3}{2}$
Write each ratio as a fraction: ⓐ $\ 27\ \text{to}\ 72\$ⓑ $\ 51\ \text{to}\ 34.$
Onthul het antwoord
1. ⓐ $\ \frac{3}{8}$
2. ⓑ $\ \frac{3}{2}$

Write each ratio as a fraction of whole numbers:

ⓐ $\ 4.8\ \text{to}\ 11.2$
ⓑ $\ 2.7\ \text{to}\ 0.54$

Onthul het antwoord

ⓐ $\ \text{4.8 to 11.2}$
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 $4.8\ \text{to}\ 11.2$ 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. $2.7\ \text{to}\ 0.54$
Write as a fraction.	$\frac{2.7}{0.54}$
Move both decimals right two places.	$\frac{270}{54}$
Simplify.	$\frac{5}{1}$

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

Write each ratio as a fraction: ⓐ $\ 4.6\ \text{to}\ 11.5\$ⓑ $\ 2.3\ \text{to}\ 0.69.$
Onthul het antwoord
1. ⓐ $\ \frac{2}{5}\$
2. ⓑ $\ \frac{10}{3}$
Write each ratio as a fraction: ⓐ $\ 3.4\ \text{to}\ 15.3\$ⓑ $\ 3.4\ \text{to}\ 0.68.$
Onthul het antwoord
1. ⓐ $\ \frac{2}{9}$
2. ⓑ $\ \frac{5}{1}$

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

Onthul het antwoord

	$1\frac{1}{4}\ \text{to}\ 2\frac{3}{8}$
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.	$\frac{10}{19}$

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

Onthul het antwoord

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

Onthul het antwoord

$\frac{9}{22}$
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?

Onthul het antwoord

ⓐ 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. $\frac{\text{total cholesterol}}{\text{HDL cholesterol}}$
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.
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.

Onthul het antwoord

$\frac{37}{8}$
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.

Onthul het antwoord

$\frac{102}{19}$

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?

Onthul het antwoord

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.	$\frac{\text{rise}}{\text{run}}$
Substitute in the given values.	$\frac{\text{1 inch}}{\text{1 foot}}$
Convert 1 foot to inches.	$\frac{\text{1 inch}}{\text{12 inches}}$
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.

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

Onthul het antwoord

$\frac{8}{3}$
Find the ratio of the first length to the second length: $1$ foot to $54$ inches.

Onthul het antwoord

$\frac{2}{9}$

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

Onthul het antwoord

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

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

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

Onthul het antwoord

$\frac{\text{123 miles}}{\text{2 hours}}$
Write the rate as a fraction: $242$ miles in $6$ hours.

Onthul het antwoord

$\frac{\text{121 miles}}{\text{3 hours}}$

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

Onthul het antwoord

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

Anita’s hourly pay rate is $\text{\$12}$ per hour.

Find the unit rate: $\text{\$630}$ for $35$ hours.

Onthul het antwoord

$18.00/hour
Find the unit rate: $\text{\$684}$ for $36$ hours.

Onthul het antwoord

$19.00/hour
Sven drives his car $455$ miles, using $14$ gallons of gasoline. How many miles per gallon does his car get?

Onthul het antwoord

Start with a rate of miles to gallons. Then divide.
$\text{455 miles to 14 gallons of gas}$
Write as a rate. $\frac{\text{455 miles}}{\text{14 gallons}}$
Divide 455 by 14 to get the unit rate. $\frac{\text{32.5 miles}}{\text{1 gallon}}$
Sven’s car gets $32.5$ miles/gallon, or $32.5$ mpg.
Find the unit rate: $423$ miles to $18$ gallons of gas.

Onthul het antwoord

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

Onthul het antwoord

28 mpg
The grocery store charges $\text{\$3.99}$ for a case of $24$ bottles of water. What is the unit price?

Onthul het antwoord

What are we asked to find? We are asked to find the unit price, which is the price per bottle.
Write as a rate. $\frac{\$3.99}{\text{24 bottles}}$
Divide to find the unit price. $\frac{\$0.16625}{\text{1 bottle}}$
Round the result to the nearest penny. $\frac{\$0.17}{\text{1 bottle}}$
The unit price is approximately $\text{\$0.17}$ per bottle. Each bottle costs about $\text{\$0.17}.$
Find the unit price. Round your answer to the nearest cent if necessary.
$\text{24-pack}$ of juice boxes for $\text{\$6.99}$

Onthul het antwoord

$0.29/box
Find the unit price. Round your answer to the nearest cent if necessary.
$\text{24-pack}$ of bottles of ice tea for $\text{\$12.72}$

Onthul het antwoord

$0.53/bottle

Which detergent has the lowest cost per load?

Onthul het antwoord

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

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

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand A Storage Bags, $\text{\$4.59}$ for $40$ count, or Brand B Storage Bags, $\text{\$3.99}$ for $30$ count

Onthul het antwoord

Brand A costs $0.11 per bag. Brand B costs $0.13 per bag. Brand A is the better buy.
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand C Chicken Noodle Soup, $\text{\$1.89}$ for $26$ ounces, or Brand D Chicken Noodle Soup, $\text{\$0.95}$ for $10.75$ ounces

Onthul het antwoord

Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy.
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
Onthul het antwoord

ⓐ
$\text{427 miles per}\ h\ \text{hours}$
Write as a rate. $\frac{\text{427 miles}}{h\ \text{hours}}$
ⓑ
$x\ \text{students to 3 teachers}$
Write as a rate. $\frac{x\ \text{students}}{\text{3 teachers}}$
ⓒ
$y\ \text{dollars for 18 hours}$
Write as a rate. $\frac{\$y}{\text{18 hours}}$
Translate the word phrase into an algebraic expression.
ⓐ $\ 689$ miles per $h$ hours ⓑ $y$ parents to $22$ students ⓒ $d$ dollars for $9$ minutes
Onthul het antwoord
1. ⓐ 689 mi/h hours
2. ⓑ y parents/22 students
3. ⓒ $d/9 min
Translate the word phrase into an algebraic expression.
ⓐ $m$ miles per $9$ hours ⓑ $x$ students to $8$ buses ⓒ $y$ dollars for $40$ hours
Onthul het antwoord
1. ⓐ m mi/9 h
2. ⓑ x students/8 buses
3. ⓒ $y/40 h
$20$ to $36$

Onthul het antwoord

$\frac{5}{9}$
$20$ to $32$
$42$ to $48$

Onthul het antwoord

$\frac{7}{8}$
$45$ to $54$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Ratios and Rate

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

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Probeer je eigen

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Meer in Arithmetic

Order of operations Fractions Percentages Powers and roots

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

Example	Rate	Write	Abbreviate	Read
\(68\) miles in \(1\) hour	\(\frac{\text{68 miles}}{\text{1 hour}}\)	\(68\) miles/hour	\(68\) mph	\(\text{68 miles per hour}\)
\(36\) miles to \(1\) gallon	\(\frac{\text{36 miles}}{\text{1 gallon}}\)	\(36\) miles/gallon	\(36\) mpg	\(\text{36 miles per gallon}\)

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

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

ⓐ \(\ \text{4.8 to 11.2}\)
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.	\(\frac{3}{7}\)

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

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

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

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

ⓐ
	\(\text{427 miles per}\ h\ \text{hours}\)
Write as a rate.	\(\frac{\text{427 miles}}{h\ \text{hours}}\)

ⓑ
	\(x\ \text{students to 3 teachers}\)
Write as a rate.	\(\frac{x\ \text{students}}{\text{3 teachers}}\)

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

ⓑ 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. \(2.7\ \text{to}\ 0.54\)
Write as a fraction.	\(\frac{2.7}{0.54}\)
Move both decimals right two places.	\(\frac{270}{54}\)
Simplify.	\(\frac{5}{1}\)

	\(1\frac{1}{4}\ \text{to}\ 2\frac{3}{8}\)
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.	\(\frac{10}{19}\)

Write as a fraction.	\(\frac{\text{total cholesterol}}{\text{HDL cholesterol}}\)
Substitute the values.	\(\frac{249}{39}\)
Simplify.	\(\frac{83}{13}\)

	Ratio of the rise to the run
Write the ratio as a fraction.	\(\frac{\text{rise}}{\text{run}}\)
Substitute in the given values.	\(\frac{\text{1 inch}}{\text{1 foot}}\)
Convert 1 foot to inches.	\(\frac{\text{1 inch}}{\text{12 inches}}\)
Simplify, dividing out common factors and units.	\(\frac{1}{12}\)