Math Core

A recipe asks for 2 quarts of broth, but your measuring cup only shows cups. A map says the park is 3 kilometers away, and you want to know how many meters that is. To answer questions like these, you need to convert from one unit to another in the same system.

Two measurement systems

In the United States, people use customary units like inches, pounds and gallons. Most of the world, and every science lab, uses metric units like meters, grams and liters. You'll convert inside one system at a time.

Customary units

Length	Weight	Capacity
1 foot (ft) = 12 inches (in)	1 pound (lb) = 16 ounces (oz)	1 cup (c) = 8 fluid ounces (fl oz)
1 yard (yd) = 3 feet = 36 inches	1 ton = 2,000 pounds	1 pint (pt) = 2 cups
1 mile (mi) = 5,280 feet		1 quart (qt) = 2 pints
		1 gallon (gal) = 4 quarts

Metric units

Length	Mass	Capacity
1 kilometer (km) = 1,000 meters (m)	1 kilogram (kg) = 1,000 grams (g)	1 liter (L) = 1,000 milliliters (mL)
1 meter = 100 centimeters (cm)
1 meter = 1,000 millimeters (mm)
1 centimeter = 10 millimeters

Metric units are all based on powers of ten. That makes metric conversions quick: you just move the decimal point.

Multiply or divide?

Think about the size of the units.

A big unit to a small unit: each big unit holds many small units, so you get more of them. Multiply.
A small unit to a big unit: it takes many small units to make one big unit, so you get fewer. Divide.

Converting units

Big unit → small unit: multiply by how many small units are in one big unit.
Small unit → big unit: divide by that same number.

Then ask: "Should my answer be a bigger or a smaller number?" If it doesn't match, switch operations.

Worked example: Big to small

How many inches are in 4 yards?

A yard is bigger than an inch, so the answer should be a bigger number. Multiply.

One yard is 36 inches, so $4 \times 36 = 144$ .

4 yards = 144 inches.

Worked example: Small to big, with a leftover

A board is 90 inches long. How long is it in feet?

An inch is smaller than a foot, so the answer should be a smaller number. Divide by 12.

$90 \div 12 = 7$ R $6$ . That is 7 feet and 6 inches left over.

You can also write the leftover as a fraction of a foot: $\dfrac{6}{12} = \dfrac{1}{2}$ . So the board is $7\tfrac{1}{2}$ feet long.

90 inches = 7 ft 6 in = $7\tfrac{1}{2}$ ft.

Worked example: Metric conversions with decimals

Change 3.5 kilometers to meters.
Change 250 centimeters to meters.

Solutions.

Kilometers are bigger, so multiply by 1,000. Multiplying by 1,000 moves the decimal point 3 places to the right: $3.5 \times 1{,}000 = 3{,}500$ . 3.5 km = 3,500 m.
Centimeters are smaller, so divide by 100. Dividing by 100 moves the decimal point 2 places to the left: $250 \div 100 = 2.5$ . 250 cm = 2.5 m.

Worked example: A two-step word problem

Mr. Lee makes 3 gallons of punch. He pours it into cups that each hold 1 cup of punch. How many cups can he fill?

There is no single fact for gallons to cups, so go one step at a time:

\begin{aligned} 3 \text{ gal} &= 3 \times 4 = 12 \text{ qt} \\ 12 \text{ qt} &= 12 \times 2 = 24 \text{ pt} \\ 24 \text{ pt} &= 24 \times 2 = 48 \text{ c} \end{aligned}

He can fill 48 cups.

Common mistake

The most common mistake is doing the wrong operation. For example, changing 5 feet to inches by dividing, $5 \div 12$ , gives a tiny number. But 5 feet is a lot of inches! Big unit to small unit means multiply: $5 \times 12 = 60$ inches.

Tip

To compare two amounts in different units, convert them to the same unit first. Is 7 cups more than 2 quarts? Two quarts is $2 \times 4 = 8$ cups, so no: 7 cups is less.

Practice

Practice 1

How many inches are in 6 feet?

Type a number.

Practice 2

How many quarts are in 5 gallons?

Type a number.

Practice 3

Change 2.4 kilograms to grams.

Type a number.

Practice 4

Change 450 centimeters to meters.

Type a number.

Practice 5

A bag of apples weighs 52 ounces. How many pounds is that? (You can type a mixed number like 2 1/2 or a decimal.)

Type a number.

Practice 6

Rosa has 3 yards of ribbon. She cuts it into pieces that are each 9 inches long. How many pieces does she get?

Type a number.

Practice 7

Which amount is the greatest?

Practice 8

Maya runs 1.2 kilometers every day from Monday to Friday. On Saturday she runs 850 meters. How many meters does she run in all six days?

Type a number.