Chapter 10 · Measurement and data

Convert Customary Units of Length, Weight and Capacity

Change one customary unit into another by multiplying or dividing. Use conversion tables and compare measurements that are in different units.

15 min read

You can name the same amount in more than one way. A door that is $7$ feet tall is also $84$ inches tall. The door did not change. Only the unit changed. A unit is the size you measure with, such as a foot or an inch.

To convert a measurement means to write the same amount with a different unit. In this lesson you convert customary units. These are the units most people in the United States use every day, such as inches, feet, pounds and gallons.

Start with length. The table below shows how the customary units of length fit together. Each row shows one length written two ways.

Customary units of length

Larger unit	Same length in a smaller unit
$1$ foot (ft)	$12$ inches (in)
$1$ yard (yd)	$3$ feet
$1$ yard (yd)	$36$ inches
$1$ mile (mi)	$5{,}280$ feet
$1$ mile (mi)	$1{,}760$ yards

Why does the number change when the unit changes? Go back to the door. An inch is smaller than a foot, so it takes more inches than feet to reach the top.

There are $12$ inches in each foot. So in $7$ feet there are $7$ groups of $12$ inches:

$7 \times 12 = 84$ , so $7$ feet $= 84$ inches.

Now go the other way. Start with $84$ inches and pack them into groups of $12$ . Each group makes $1$ foot:

$84 \div 12 = 7$ , so $84$ inches $= 7$ feet.

Both number sentences use $12$ , the number of inches in $1$ foot. You multiply by it to change to the smaller unit. You divide by it to change to the larger unit.

Key concept

Multiply for a smaller unit, divide for a larger unit

First find the fact that links the two units, such as $1$ foot $= 12$ inches. Then ask: is the new unit smaller or larger?

Changing to a smaller unit: you need more of them, so multiply by the number in the fact.
Changing to a larger unit: you need fewer of them, so divide by the number in the fact.

The amount stays the same. Only the unit changes, and the number changes with it.

Worked example

Miles to feet

A hiking trail is $3$ miles long. How many feet long is it?

Find the fact that links miles and feet. The length table says that 1 mile is the same length as 5,280 feet.
$1 \text{ mile} = 5{,}280 \text{ feet}$
Decide whether to multiply or divide. A foot is much smaller than a mile, so it takes more feet than miles to cover the trail. More means multiply.
Multiply the number of miles by 5,280, because each mile is 5,280 feet and there are 3 miles. To make it easier, split 5,280 into 5,000 and 280. Multiply each part by 3, then add.
$3 \times 5{,}280 = (3 \times 5{,}000) + (3 \times 280) = 15{,}000 + 840 = 15{,}840$
Check that the answer fits the rule. You changed to a smaller unit, so the number should go up. It did: 15,840 is much more than 3.

Answer

The hiking trail is $15{,}840$ feet long.

The same rule works for two other kinds of measurement.

Weight tells how heavy something is. Its customary units are ounces, pounds and tons.

Capacity tells how much a container can hold. Its customary units are fluid ounces, cups, pints, quarts and gallons. A fluid ounce measures capacity, so it is a different unit from the ounce that measures weight.

The table below gives the facts that link these units. Read it the same way as the length table. Each row shows one amount written two ways.

Customary units of weight and capacity

Kind of measurement	Larger unit	Same amount in a smaller unit
Weight	$1$ pound (lb)	$16$ ounces (oz)
Weight	$1$ ton (T)	$2{,}000$ pounds
Capacity	$1$ cup (c)	$8$ fluid ounces (fl oz)
Capacity	$1$ pint (pt)	$2$ cups
Capacity	$1$ quart (qt)	$2$ pints
Capacity	$1$ gallon (gal)	$4$ quarts

Worked example

Pints to gallons

A cooler holds $40$ pints of water. How many gallons does it hold?

Look in the table for a fact that links pints and gallons. No single row does that. But one row links pints to quarts, and another links quarts to gallons. So convert in two moves: first pints to quarts, then quarts to gallons.
Change pints to quarts with the fact 1 quart = 2 pints. A quart is larger than a pint, so you need fewer quarts. Divide by 2, because every 2 pints make 1 quart. So 40 pints is 20 quarts.
$40 \div 2 = 20$
Change quarts to gallons with the fact 1 gallon = 4 quarts. A gallon is larger than a quart, so divide again. Every 4 quarts make 1 gallon. So 20 quarts is 5 gallons.
$20 \div 4 = 5$
Check by multiplying your way back. 5 gallons is 5 groups of 4 quarts, which is 20 quarts. 20 quarts is 20 groups of 2 pints, which is 40 pints. That matches the cooler.
$5 \times 4 = 20 \qquad 20 \times 2 = 40$

Answer

The cooler holds $5$ gallons of water.

A conversion table lists matching amounts in two units. Each row shows one amount written two ways, so the two numbers in a row stand for equal amounts.

Here is a conversion table for pounds and ounces. Two numbers are missing. Before you read on, look across the rows that are filled in. What happens to the pounds number to make the ounces number?

Conversion table for pounds and ounces

Pounds	Ounces
$1$	$16$
$2$	$32$
$3$	$48$
$4$	$64$
$5$	?
?	$96$

Look across each filled row and write what you see as a number sentence.

Row 1: $1$ pound is $16$ ounces, and $1 \times 16 = 16$ .
Row 2: $2$ pounds is $32$ ounces, and $2 \times 16 = 32$ .
Row 3: $3$ pounds is $48$ ounces, and $3 \times 16 = 48$ .

The pattern between the two columns is the same in every row: the number of ounces is $16$ times the number of pounds. This makes sense, because there are $16$ ounces in each pound. Going the other way, the number of pounds is the number of ounces divided by $16$ .

Use the pattern to complete the table.

$5$ pounds: multiply. $5 \times 16 = 80$ , so the missing number is $80$ ounces.
$96$ ounces: divide. $96 \div 16 = 6$ , so the missing number is $6$ pounds.

You can also use the table to look up an answer. How many pounds is $64$ ounces? Find $64$ in the ounces column and read across the row. It is $4$ pounds.

Worked example

Which load is heavier?

A load of sand weighs $3$ tons. A load of gravel weighs $5{,}500$ pounds. Which load is heavier?

You cannot compare 3 and 5,500 yet. One number counts tons and the other counts pounds, and those units are different sizes. First write both weights in the same unit. You can convert either one. Changing tons to pounds is the easier choice here, because you multiply.
Use the fact 1 ton = 2,000 pounds. A pound is smaller than a ton, so multiply. Each ton is 2,000 pounds, so 3 tons is 3 groups of 2,000 pounds. The sand weighs 6,000 pounds.
$3 \times 2{,}000 = 6{,}000$
Now both weights are in pounds, so you can compare the numbers. The sand is 6,000 pounds and the gravel is 5,500 pounds.
$6{,}000 > 5{,}500$
Write the comparison with the units from the problem. 6,000 pounds is the same weight as 3 tons, so 3 tons is more than 5,500 pounds.
$3 \text{ tons} > 5{,}500 \text{ pounds}$

Answer

The load of sand is heavier. It weighs $3$ tons, which is $6{,}000$ pounds, and that is more than $5{,}500$ pounds.

Here is the idea to keep. Converting gives an amount a new name, but the amount stays the same. Find the fact that links the two units. Then multiply to change to a smaller unit, because you need more of them. Divide to change to a larger unit, because you need fewer.

A conversion table shows that same pattern in every row. To compare two measurements, convert one of them so both use the same unit. Then compare the numbers.

Next, you will convert measurements that have fractions and mixed numbers. You will also add and subtract measurements given in feet and inches, and solve word problems with more than one step.

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Customary Units: Fractions, Mixed Units and Problems

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