Converting Between Customary and Metric Units | Grade 7 Math | LevelKite

A water bottle label might say 1 liter, while a milk jug says 1 gallon. Those labels come from two measurement systems. U.S. customary units include inches, feet, miles, ounces, pounds, quarts and gallons. Metric units include centimeters, meters, kilometers, grams, kilograms, milliliters and liters.

You already know how to convert inside the metric system. In this lesson you will cross from one system to the other, and then use that skill to solve word problems.

Keep one idea in mind the whole time: converting never changes the amount. A doorway is the same height whether you measure it in inches or in centimeters. Only the unit changes, and the number changes with it so that the amount stays the same.

The bridge between the two systems is a conversion fact: a statement that two measurements name the same amount. For example, $1\text{ in} = 2.54\text{ cm}$ says that $1$ inch and $2.54$ centimeters are the same length.

You can read a conversion fact as a rate, which compares two quantities that have different units: $2.54$ centimeters per $1$ inch. It is a unit rate because the quantity after per, called the per-one quantity, is exactly $1$ . Written as a division, it is $2.54\text{ cm} \div 1\text{ in}$ . The per-one quantity is the divisor, and a divisor is never zero.

Every inch matches another $2.54$ centimeters, so $2$ inches is $5.08$ centimeters and $3$ inches is $7.62$ centimeters. The number of centimeters is always $2.54$ times the number of inches. Every measurement in this lesson is a positive quantity: a length, a weight, or a capacity (how much a container holds).

Most conversion facts are rounded. For those, write $\approx$ , which means is about equal to, instead of $=$ .

Common conversion facts. Only the inch fact is exact; the others are rounded.

What it measures	Customary unit	Metric unit	Conversion fact	Larger unit
Length	inch (in)	centimeter (cm)	$1\text{ in} = 2.54\text{ cm}$	inch
Length	foot (ft)	meter (m)	$1\text{ ft} \approx 0.305\text{ m}$	meter
Length	mile (mi)	kilometer (km)	$1\text{ mi} \approx 1.61\text{ km}$	mile
Weight (mass)	ounce (oz)	gram (g)	$1\text{ oz} \approx 28.35\text{ g}$	ounce
Weight (mass)	pound (lb)	kilogram (kg)	$1\text{ lb} \approx 0.454\text{ kg}$	kilogram
Capacity	gallon (gal)	liter (L)	$1\text{ gal} \approx 3.79\text{ L}$	gallon

Key concept

Multiply by a conversion factor equal to 1

A conversion factor is a fraction made from one conversion fact, with the same amount on top and on the bottom. The numbers differ, but $2.54\text{ cm}$ and $1\text{ in}$ are the same length, so $\frac{2.54\text{ cm}}{1\text{ in}} = 1$ , just as $\frac{100\text{ cm}}{1\text{ m}} = 1$ .

Multiplying by $1$ never changes an amount, so multiplying a measurement by a conversion factor changes only its unit and its number.

This works when:

the measurements are positive lengths, weights or capacities;
every number keeps its unit;
the unit you are leaving is in the denominator, where it matches and cancels, and the denominator is not zero.

When the fact is rounded, the answer is rounded too, so write $\approx$ .

\text{measurement in old unit} \times \dfrac{\text{amount in new unit}}{\text{equal amount in old unit}} = \text{measurement in new unit}

Worked example

Inches to centimeters

Convert $15$ inches to centimeters.

Pick the conversion fact that links the two units. Inches and centimeters both measure length, and 1 inch is exactly 2.54 centimeters.
$1\text{ in} = 2.54\text{ cm}$
Build the conversion factor. You are leaving inches, so inches go in the denominator. The top and the bottom are the same length, so the factor equals 1. Its denominator, 1 inch, is not zero.
$\dfrac{2.54\text{ cm}}{1\text{ in}} = 1$
Multiply 15 inches by the factor. Multiplying by 1 keeps the length the same. Inches are on top once and on the bottom once, so they cancel and leave centimeters.
$15\text{ in} \times \dfrac{2.54\text{ cm}}{1\text{ in}} = (15 \times 2.54)\text{ cm}$
Multiply the numbers. Split 15 into 10 and 5, multiply each part by 2.54, and add the two products.
$15 \times 2.54 = (10 \times 2.54) + (5 \times 2.54) = 25.4 + 12.7 = 38.1$
Check the size. A centimeter is shorter than an inch, so the same positive length needs more centimeters than inches. The result 38.1 is greater than 15, as it should be.
$38.1 > 15$

Answer

$15$ inches is exactly $38.1$ centimeters.

In the first example you multiplied by the number in the fact. Going the other way, from metric to customary, uses the same idea. One conversion fact gives two conversion factors, because either amount can go on top. Each factor has the same amount on top and on the bottom, so each is equal to $1$ , or about $1$ when the fact is rounded.

Choose the factor whose denominator has the unit you are leaving. That unit matches the unit on your measurement and cancels, and the unit you want stays on top. Here are the two factors for miles and kilometers:

\dfrac{1.61\text{ km}}{1\text{ mi}} \approx 1 \qquad \text{and} \qquad \dfrac{1\text{ mi}}{1.61\text{ km}} \approx 1

The first factor changes miles to kilometers. The second changes kilometers to miles.

You can also write a conversion as a proportion, which is an equation saying that two ratios are equal. A ratio compares two quantities by division. When you set up a proportion, keep corresponding quantities in matching positions: if miles are on top in one ratio, miles must be on top in the other, with kilometers on the bottom in both. Otherwise the two ratios compare different things, and you cannot set them equal. The next example starts with a proportion and ends with the same factor.

Worked example

Kilometers to miles

A 5K race is $5$ kilometers long. Convert $5$ kilometers to miles. Use $1\text{ mi} \approx 1.61\text{ km}$ and round to the nearest tenth of a mile.

Name the unknown and its unit. Let m be the number of miles in 5 kilometers. A distance is positive, so m is greater than 0.
$m = \text{number of miles in }5\text{ km}, \quad m > 0$
Write a proportion. Both ratios compare miles to kilometers, so miles go on top and kilometers go on the bottom in each ratio. The distances m miles and 5 kilometers are the same, and 1 mile and 1.61 kilometers are about the same, so both ratios are about 1.
$\dfrac{m\text{ mi}}{5\text{ km}} \approx \dfrac{1\text{ mi}}{1.61\text{ km}}$
Multiply both sides by 5 kilometers. Multiplying both sides by the same nonzero amount keeps the proportion true. On the left, 5 kilometers cancels. The right side becomes 5 kilometers times the factor with kilometers in its denominator, so kilometers cancel and miles are left.
$m\text{ mi} \approx 5\text{ km} \times \dfrac{1\text{ mi}}{1.61\text{ km}} = \dfrac{5}{1.61}\text{ mi}$
Divide. First multiply both numbers by 100, which makes the divisor a whole number and does not change the quotient. Then round to the nearest tenth: the hundredths digit is 0, which is less than 5, so the tenths digit stays 1.
$5 \div 1.61 = 500 \div 161 = 3.105\ldots \approx 3.1$
Check two ways. Convert back with the other factor and see that you land close to 5 kilometers. Then check the size: a mile is longer than a kilometer, so a positive distance has fewer miles than kilometers, and 3.1 is less than 5.
$3.1\text{ mi} \times \dfrac{1.61\text{ km}}{1\text{ mi}} = 4.991\text{ km} \approx 5\text{ km}$

Answer

$5$ kilometers is about $3.1$ miles.

Word problems rarely say convert. They give measurements in different units and ask a question about them. You can add, subtract or compare two measurements of the same kind only when both are written in the same unit. The same is true when you ask how many times one amount fits into another.

For example, which is longer, a $6$ -mile trail or a $10$ -kilometer trail? You cannot compare $6$ and $10$ until the units match. In kilometers, the first trail is about $6\text{ mi} \times \frac{1.61\text{ km}}{1\text{ mi}} = 9.66\text{ km}$ , so the $10$ -kilometer trail is a little longer.

A plan that works:

Find what the question asks for, and in which unit.
Write each measurement with its unit. Convert until measurements of the same kind share one unit.
Do the rest of the mathematics with the units still attached.
Check the size of your result, then answer in a sentence.

Worked example

Filling water bottles

A team cooler holds $5$ gallons of water. Each player's bottle holds $750$ milliliters. How many bottles can the team fill completely from one full cooler? Use $1\text{ gal} \approx 3.79\text{ L}$ .

Find what the question asks for. It asks for a number of bottles, so you will divide the cooler's capacity by one bottle's capacity. Gallons and milliliters do not match, so first write both capacities in liters.
Convert the cooler to liters. You are leaving gallons, so gallons go in the denominator and cancel. The fact is rounded, so the cooler holds about 18.95 liters.
$5\text{ gal} \times \dfrac{3.79\text{ L}}{1\text{ gal}} = 18.95\text{ L}$
Convert one bottle to liters with a metric fact you already know: 1000 milliliters is 1 liter. Milliliters go in the denominator and cancel.
$750\text{ mL} \times \dfrac{1\text{ L}}{1000\text{ mL}} = 0.75\text{ L}$
Divide the cooler's liters by the liters in one bottle. Each bottle holds 0.75 liter, a unit rate of 0.75 liter per 1 bottle. This divisor is positive, so it is not zero, and liters cancel to leave bottles.
$18.95\text{ L} \div \dfrac{0.75\text{ L}}{1\text{ bottle}} = (18.95 \div 0.75)\text{ bottles} = 25.266\ldots\text{ bottles}$
Interpret the quotient. Only completely filled bottles count, so keep the whole-number part, 25. Check by multiplying: 25 bottles need 18.75 liters, which the cooler has, but 26 bottles need 19.5 liters, which is more than 18.95 liters.
$25 \times 0.75 = 18.75 < 18.95 \quad \text{and} \quad 26 \times 0.75 = 19.5 > 18.95$

Answer

The team can completely fill $25$ bottles, and about $0.2$ liter of water is left in the cooler.

The big idea: converting between customary and metric units changes the unit and the number, but never the amount. A conversion fact gives you a factor equal to $1$ , or about $1$ if the fact is rounded. Put the unit you are leaving in the denominator so that it matches and cancels, and keep a unit beside every number. For positive measurements, check the size: a larger unit means a smaller number. In a word problem, make the units match first. Then finish the mathematics and answer in a sentence.