Converting Units, Including Customary and Metric | Grade 8 Math | LevelKite

A measurement is a number together with a unit, the fixed amount you count with, such as an inch or a gram. You can write the same amount with different units. A board that is $3$ feet (ft) long is also $36$ inches (in) long. The board has not changed. Only the way you name its length has.

To convert a measurement means to rewrite it with a different unit without changing the amount. The number changes because the size of the unit changes. An inch is shorter than a foot, so you need more inches than feet to cover the same positive length.

In this lesson you convert units of length, of weight or mass (how heavy something is) and of capacity (how much a container holds). For these quantities, one unit always equals a fixed positive number of another unit of the same kind. For example, one foot always equals $12$ inches. So for a positive amount, a smaller unit always gives a larger number, and a larger unit gives a smaller number.

A conversion fact is an equation that says two measurements are the same amount, such as $1\text{ ft} = 12\text{ in}$ . You can also read it as a rate, a comparison of two quantities with different units: $12$ inches per foot.

Write the two equal measurements as a fraction, $\frac{1\text{ ft}}{12\text{ in}}$ . Its numerator (the top) and its denominator (the bottom) are the same nonzero length, so the fraction equals $1$ . A fraction like this is a conversion factor. Its denominator is never zero, which matters because division by zero is undefined.

Multiplying by $1$ does not change an amount. So when you multiply a measurement by a conversion factor, the amount stays the same. Only the number and the unit change. A unit that appears in both a numerator and a denominator cancels: the matching units divide to make $1$ , just as a nonzero common factor does. Only matching units cancel. Inches cancel inches, never feet.

Here is how you convert $36$ inches to feet:

36\text{ in} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{36}{12}\text{ ft} = 3\text{ ft}

Key concept

Multiply by a conversion factor equal to 1

To convert a measurement, multiply it by a conversion factor: a fraction made from two equal, nonzero measurements, so its value is $1$ . Put the unit you want in the numerator and the unit you want to remove in the denominator. The matching units cancel, and the amount stays the same because you multiplied by $1$ .

This works when both units measure the same kind of quantity and one unit equals a fixed positive number of the other, as with the units of length, weight or mass, and capacity in this lesson.

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

Units come in two main families. Customary units are the everyday units of the United States, such as inches (in), feet (ft), yards (yd), miles (mi), ounces (oz), pounds (lb), quarts (qt) and gallons (gal). Metric units are related by powers of ten such as $100$ and $1{,}000$ . They include centimeters (cm), meters (m), kilometers (km), grams (g), kilograms (kg), milliliters (mL) and liters (L).

Start inside one system. Every fact in the table below is exact: the two measurements are precisely the same amount, with nothing rounded.

Each fact gives you two conversion factors, such as $\frac{1\text{ ft}}{12\text{ in}}$ and its reciprocal (the same fraction turned upside down), $\frac{12\text{ in}}{1\text{ ft}}$ . Both equal $1$ , because the two measurements are equal and neither is zero. Choose the one whose denominator matches the unit you want to remove.

Exact conversion facts inside one system

System	Quantity	Conversion fact
Customary	Length	$1\text{ ft} = 12\text{ in}$
Customary	Length	$1\text{ yd} = 3\text{ ft}$
Customary	Length	$1\text{ mi} = 5{,}280\text{ ft}$
Customary	Weight or mass	$1\text{ lb} = 16\text{ oz}$
Customary	Capacity	$1\text{ gal} = 4\text{ qt}$
Metric	Length	$1\text{ m} = 100\text{ cm}$
Metric	Length	$1\text{ km} = 1{,}000\text{ m}$
Metric	Weight or mass	$1\text{ kg} = 1{,}000\text{ g}$
Metric	Capacity	$1\text{ L} = 1{,}000\text{ mL}$

Worked example

One step inside one system

A trail is $7{,}920$ feet long. How many miles long is it?

Find the fact that links the two units. Feet and miles are both customary units of length, and the table gives an exact fact for them.
$1\text{ mi} = 5{,}280\text{ ft}$
Choose the conversion factor that removes feet. Feet must be in the denominator so they match the feet in the measurement and cancel. The factor equals 1 because its numerator and its nonzero denominator are the same length.
$\frac{1\text{ mi}}{5{,}280\text{ ft}} = 1$
Multiply the measurement by the factor. The feet cancel and miles are left. The length stays the same because you multiplied by 1.
$7{,}920\text{ ft} \times \frac{1\text{ mi}}{5{,}280\text{ ft}} = \frac{7{,}920}{5{,}280}\text{ mi}$
Simplify the fraction. Divide the numerator and the denominator by 10, then by 8, then by 33. Dividing both by the same nonzero number keeps the value of the fraction the same.
$\frac{7{,}920}{5{,}280} = \frac{792}{528} = \frac{99}{66} = \frac{3}{2} = 1.5$
Check the size. A mile is much longer than a foot, so a positive length needs far fewer miles than feet. The result 1.5 is far less than 7,920, as it should be.

Answer

The trail is exactly $1.5$ miles long.

The same method takes a measurement from one system to the other. You only need a conversion fact that links a customary unit to a metric unit.

In the table below, one fact is exact: by definition, one inch is exactly $2.54$ centimeters. The other facts are rounded, which means the true number has more decimal places than the table shows. The symbol $\approx$ means is approximately equal to.

A conversion factor made from a rounded fact is very close to $1$ , not exactly $1$ , so it gives an amount very close to the original. When every fact you use is exact and you round nothing in your arithmetic, your answer is exact. When you use a rounded fact or round your arithmetic, treat your answer as an approximation, a value close to the true one, and write it with $\approx$ .

Conversion facts that link customary and metric units

Quantity	Conversion fact	Exact or rounded
Length	$1\text{ in} = 2.54\text{ cm}$	Exact
Length	$1\text{ mi} \approx 1.609\text{ km}$	Rounded
Weight or mass	$1\text{ kg} \approx 2.205\text{ lb}$	Rounded
Weight or mass	$1\text{ oz} \approx 28.35\text{ g}$	Rounded
Capacity	$1\text{ gal} \approx 3.785\text{ L}$	Rounded
Capacity	$1\text{ L} \approx 1.057\text{ qt}$	Rounded

Worked example

One step between systems

A fish tank holds $2.5$ gallons of water. About how many liters does it hold? Round to the nearest tenth of a liter.

Find the fact that links gallons and liters. It is a rounded fact, so treat the result as an approximation.
$1\text{ gal} \approx 3.785\text{ L}$
Put gallons in the denominator so they match the gallons in the measurement and cancel. The denominator is not zero. Because the fact is rounded, the factor is very close to 1, so the amount stays very nearly the same.
$2.5\text{ gal} \times \frac{3.785\text{ L}}{1\text{ gal}} = (2.5 \times 3.785)\text{ L}$
Multiply the numbers. Because 2.5 is 2 plus 0.5, you can multiply each part by 3.785 and add the two results.
$2.5 \times 3.785 = 2 \times 3.785 + 0.5 \times 3.785 = 7.57 + 1.8925 = 9.4625$
Round 9.4625 to the nearest tenth. The hundredths digit is 6, which is 5 or more, so the tenths digit goes up from 4 to 5. Then check the size: a liter is smaller than a gallon, so a positive amount needs more liters than gallons, and 9.5 is more than 2.5.
$2.5\text{ gal} \approx 9.5\text{ L}$

Answer

The tank holds about $9.5$ liters of water.

Watch for this mistake: multiplying by the number in a conversion fact without checking the units.

Suppose you convert $60$ cm to inches by writing $60 \times 2.54 = 152.4$ inches. That multiplies by $\frac{2.54\text{ cm}}{1\text{ in}}$ , so the centimeters do not cancel. The units left are centimeters times centimeters per inch, not inches. The size is wrong too. An inch is longer than a centimeter, so a positive length needs fewer inches than centimeters, yet $152.4$ is more than $60$ .

Put centimeters in the denominator so they cancel: $60\text{ cm} \times \frac{1\text{ in}}{2.54\text{ cm}} = \frac{60}{2.54}\text{ in} \approx 23.6\text{ in}$ , rounded to the nearest tenth. As a check, $23.6 \times 2.54 = 59.944$ , which is close to $60$ .

Worked example

Two steps in a chain

A rope is $381$ centimeters long. How many feet long is it?

Plan a path. No fact in the tables links centimeters straight to feet, so go through inches: centimeters to inches, then inches to feet. Both facts are exact.
$1\text{ in} = 2.54\text{ cm} \quad\text{and}\quad 1\text{ ft} = 12\text{ in}$
Multiply by one factor for each link, with the unit you want to remove in each denominator. Each factor equals 1 because its two measurements are equal and not zero, so the length stays the same. Centimeters cancel centimeters, inches cancel inches, and feet are left.
$381\text{ cm} \times \frac{1\text{ in}}{2.54\text{ cm}} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{381}{2.54 \times 12}\text{ ft}$
Dividing by 2.54 times 12 is the same as dividing by 2.54 and then by 12. Start with 2.54. Multiply 381 and 2.54 by 100 to clear the decimal; scaling both by the same nonzero number does not change the result of the division. That result is 150, because 254 times 150 is 25,400 plus 12,700, which is 38,100. So the rope is 150 inches long.
$\frac{381}{2.54} = \frac{38{,}100}{254} = 150$
Divide 150 by 12 to change inches to feet. Twelve goes into 150 twelve times, because 12 times 12 is 144, with 6 left over. Six out of 12 is 0.5.
$\frac{150}{12} = 12 + \frac{6}{12} = 12.5$
Check the size. A foot is longer than a centimeter, so a positive length needs fewer feet than centimeters, and 12.5 is less than 381. You used only exact facts and rounded nothing, so the answer is exact.

Answer

The rope is exactly $12.5$ feet long.

Converting a measurement changes its number and its unit but never its amount. You multiply by a conversion factor, a fraction of two equal, nonzero measurements, and you place the units so that matching units cancel and only the unit you want is left. When no single fact links your units, multiply by one factor for each link.

Your answer is exact when every fact you use is exact and you round nothing. If you use a rounded fact or round your arithmetic, write the answer as an approximation with $\approx$ . For a positive measurement in units like these, a larger unit means a smaller number and a smaller unit means a larger number. Use that to check your answers.

Next, you will use these same conversion factors to convert rates as well as measurements.