Math Core

A recipe calls for 3 quarts of broth, but your measuring cup only shows cups. A race is 5 kilometers, and you want to know how many meters that is. Every unit conversion is really a ratio problem, so the tools you already have will do the job.

Conversion facts are ratios

A fact like "1 foot = 12 inches" is a ratio: 1 foot for every 12 inches. You can write it as a rate either way:

\frac{12 \text{ inches}}{1 \text{ foot}} \qquad \text{or} \qquad \frac{1 \text{ foot}}{12 \text{ inches}}

Here are some common conversion facts:

Length	Weight	Capacity	Time
1 foot = 12 inches	1 pound = 16 ounces	1 cup = 8 fluid ounces	1 minute = 60 seconds
1 yard = 3 feet	1 ton = 2,000 pounds	1 pint = 2 cups	1 hour = 60 minutes
1 mile = 5,280 feet	1 kilogram = 1,000 grams	1 quart = 2 pints	1 day = 24 hours
1 meter = 100 centimeters		1 gallon = 4 quarts
1 kilometer = 1,000 meters		1 liter = 1,000 milliliters

Using a ratio table

Put the conversion fact in the first column. Then multiply or divide to reach the amount you have.

Worked example: Quarts to cups

How many cups are in 3 quarts?

1 quart = 2 pints and 1 pint = 2 cups, so 1 quart = 4 cups.

Quarts	1	3
Cups	4	12

There are 12 cups in 3 quarts.

Using a unit rate

You can also multiply by the unit rate. Choose the rate so that the unit you have is on the bottom and the unit you want is on top. Then the unit you have cancels out.

Converting with a rate

Multiply by a conversion rate written as a fraction:

\text{amount you have} \times \frac{\text{new unit}}{\text{old unit}}

The old unit cancels, and you are left with the new unit.

Worked example: Kilometers to meters

A race is 5 kilometers long. How many meters is that?

5 \text{ km} \times \frac{1{,}000 \text{ m}}{1 \text{ km}} = 5{,}000 \text{ m}

The race is 5,000 meters long.

Big units or small units?

Before you calculate, decide whether your answer should be a bigger or a smaller number.

Converting to a smaller unit (feet to inches): you need more of them, so multiply.
Converting to a larger unit (inches to feet): you need fewer of them, so divide.

Worked example: Ounces to pounds

A bag of flour weighs 80 ounces. How many pounds is that?

A pound is bigger than an ounce, so the answer should be a smaller number. Divide:

80 \text{ oz} \times \frac{1 \text{ lb}}{16 \text{ oz}} = \frac{80}{16} \text{ lb} = 5 \text{ lb}

The flour weighs 5 pounds.

Common mistake

The most common mistake is multiplying when you should divide. If you did $80 \times 16 = 1{,}280$ , you would be saying the flour weighs 1,280 pounds! Always ask: is the new unit bigger or smaller? Bigger units mean a smaller number.

Converting rates

You can convert a rate, too. Convert one unit at a time.

Worked example: Converting a speed

A snail crawls 3 feet per minute. How many feet does it crawl per hour?

There are 60 minutes in 1 hour, so in 1 hour the snail crawls 60 times as far:

\frac{3 \text{ ft}}{1 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} = \frac{180 \text{ ft}}{1 \text{ hr}}

The snail crawls 180 feet per hour.

Tip

Check your answer with a quick estimate. A foot is about the length of a notebook, so 7 feet should be a lot of inches. $7 \times 12 = 84$ inches makes sense; $7 \div 12$ , which is less than 1, does not.