Math Core

Is a 12-ounce box of cereal for $3.60 a better deal than a 20-ounce box for $5.40? How fast is a car that goes 150 miles in 2.5 hours? Questions like these compare two quantities, and the tools for them are ratios and rates. They are also the first step toward the equations you'll solve in algebra.

Ratios

A ratio compares two quantities. If a class has 12 boys and 18 girls, the ratio of boys to girls can be written three ways:

12 \text{ to } 18 \qquad 12 : 18 \qquad \frac{12}{18}

Order matters. The ratio of girls to boys is $18 : 12$ , which is a different comparison.

Definition

Ratio

A ratio $a : b$ compares a quantity $a$ to a quantity $b$ . Two ratios are equivalent if you can get one from the other by multiplying or dividing both parts by the same nonzero number.

Because a ratio can be written as a fraction, you simplify it the same way: divide both parts by their greatest common factor. The GCF of 12 and 18 is 6, so

12 : 18 = \frac{12}{18} = \frac{2}{3} = 2 : 3.

For every 2 boys there are 3 girls. You can also compare a part to the whole. There are $12 + 18 = 30$ students, so the ratio of girls to all students is $18 : 30 = 3 : 5$ .

Worked example: Using a ratio to find amounts

A paint mix uses blue and yellow in the ratio $2 : 5$ . A painter makes 28 liters of the mix. How many liters are blue?

Each "batch" of the ratio has $2 + 5 = 7$ parts. The 28 liters split into 7 equal parts:

28 \div 7 = 4 \text{ liters per part.}

Blue is 2 parts: $2 \times 4 = 8$ liters. (Yellow is $5 \times 4 = 20$ liters, and $8 + 20 = 28$ . ✓)

Rates and unit rates

A rate is a ratio of two quantities with different units, like miles per hour or dollars per pound. The most useful form is a unit rate, where the second quantity is 1.

Unit rate

To find a unit rate, divide the first quantity by the second:

\text{unit rate} = \frac{\text{amount}}{\text{1 unit of the other quantity}}

150 miles in 2.5 hours is $\dfrac{150}{2.5} = 60$ miles per hour.

Once you know a unit rate, you can find any other amount by multiplying. At 60 miles per hour, 4 hours of driving covers $60 \times 4 = 240$ miles.

Worked example: Comparing unit prices

Which cereal box is the better buy: 12 ounces for $3.60 or 20 ounces for $5.40?

Find the price per ounce for each.

Small box: $3.60 \div 12 = 0.30$ , so $0.30 per ounce.
Large box: $5.40 \div 20 = 0.27$ , so $0.27 per ounce.

The large box costs less per ounce, so it is the better buy.

Common mistake

Keep the order of the quantities consistent. "Dollars per ounce" means dollars divided by ounces. If you divide ounces by dollars for one box and dollars by ounces for the other, the comparison is meaningless. Write the units next to every number so you can see what you're dividing.

Rates with fractions

A unit rate still works when the quantities are fractions. Divide as you learned for fractions: multiply by the reciprocal.

Worked example: A unit rate from fractions

Marcus walks $\dfrac{3}{4}$ of a mile in $\dfrac{1}{5}$ of an hour. What is his speed in miles per hour?

\frac{3}{4} \div \frac{1}{5} = \frac{3}{4} \times \frac{5}{1} = \frac{15}{4} = 3\tfrac{3}{4}

He walks $3\tfrac{3}{4}$ miles per hour.

Converting units with rates

A rate like "5,280 feet per mile" equals 1, because 5,280 feet and 1 mile are the same length. Multiplying by a fraction equal to 1 changes the units without changing the amount. Set up each fraction so the unit you want to remove appears once on top and once on the bottom, then cancel it.

Worked example: Feet per second

A cyclist rides at 15 miles per hour. How many feet per second is that? (1 mile = 5,280 feet, 1 hour = 3,600 seconds.)

\frac{15 \text{ mi}}{1 \text{ h}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{15 \times 5280}{3600} \ \frac{\text{ft}}{\text{s}} = \frac{79200}{3600} = 22

Miles cancel, hours cancel, and feet per second remain: 22 feet per second.

Tip

Before you calculate, predict whether the number should get bigger or smaller. A foot is much shorter than a mile, so the number of feet should be large; a second is much shorter than an hour, so the number per second should be smaller. If your answer doesn't fit your prediction, check which way you set up each fraction.

Practice

Practice 1

A bag has 24 red marbles and 36 blue marbles. What is the ratio of red to blue in simplest form?

Practice 2

Priya types 416 words in 8 minutes. What is her unit rate in words per minute?