Compare and Convert Customary and Metric Units | Grade 7 Math | LevelKite

A unit of measurement is a fixed amount, such as an inch or a pound, that you use to say how much of something there is. The same amount can have different names. A ruler that is $1$ foot long is also $12$ inches long. The ruler does not change. Only the unit that names its length changes.

Converting a measurement means renaming it in a different unit without changing the amount. In this lesson you will convert and compare positive measurements of length, weight (how heavy an object is), mass (the amount of matter in an object) and capacity (the amount a container can hold).

Notice what happens to the numbers. A foot is longer than an inch, so the same positive length takes fewer feet ( $1$ ) than inches ( $12$ ). You will start with the customary system, the set of units such as inches, feet, pounds and gallons that is common in the United States.

Customary unit facts: each row gives two names for the same amount

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

Each row of the table lets you build a conversion factor: a ratio of two equal measurements written in different units. From the fact $1\text{ ft} = 12\text{ in}$ you can write two of them:

$\frac{12\text{ in}}{1\text{ ft}}$ and $\frac{1\text{ ft}}{12\text{ in}}$

You can read the first one as a unit rate: $12$ inches per $1$ foot. The per-one quantity, $1$ foot, is the divisor, and it is not zero. The numbers $12$ and $1$ are different, but $12$ inches and $1$ foot are the same positive length, so each factor equals $1$ . Multiplying a measurement by $1$ keeps the amount the same. Only the unit that names it changes.

To rename $3$ feet in inches, choose the factor with feet on the bottom, so that it matches the unit you are leaving. Then feet divide out (cancel), the way a common factor divides out of a fraction, and inches are left:

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

Key concept

Convert by multiplying by a factor equal to 1

To convert a positive measurement of length, weight, mass or capacity, multiply it by a conversion factor: a ratio of two equal, nonzero measurements in different units. The factor equals $1$ , so the amount stays the same and only its unit changes.

Keep the unit on every number.
Put the unit you are leaving on the bottom (the divisor) so that it matches your measurement's unit and divides out. Put the unit you want on the top.
To compare measurements, first rename them in one shared unit. Then the greater number is the greater amount.

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

Worked example

Convert inches to feet

Convert $78$ inches to feet.

Find the fact that links the two units: 1 foot and 12 inches are the same length. Neither amount is zero, so their ratio is a conversion factor equal to 1.
$1\text{ ft} = 12\text{ in}$
You are leaving inches, so choose the factor with inches on the bottom, to match the unit of 78 inches. Multiplying by this factor keeps the length the same, because the factor equals 1.
$78\text{ in} \times \frac{1\text{ ft}}{12\text{ in}}$
Inches on the top and inches on the bottom divide out, so feet are the only unit left. The numbers that remain are 78 divided by 12.
$78\text{ in} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{78}{12}\text{ ft}$
Divide 78 by 12. Six 12s make 72, which leaves 6. Then 6 out of 12 is one half, or 0.5. So 78 divided by 12 is 6.5.
$\frac{78}{12}\text{ ft} = \left(6 + \frac{6}{12}\right)\text{ ft} = 6.5\text{ ft}$
Check the result two ways. A foot is longer than an inch, so a positive length takes fewer feet than inches, and 6.5 is less than 78. Converting back with the other factor also returns the starting measurement.
$6.5\text{ ft} \times \frac{12\text{ in}}{1\text{ ft}} = (6.5 \times 12)\text{ in} = 78\text{ in}$

Answer

$78$ inches is the same length as $6.5$ feet.

Worked example

Compare gallons and pints

Which is the greater amount: $2.5$ gallons or $22$ pints?

Comparing the numbers tells you which amount is greater only when both measurements use the same unit. Rename 2.5 gallons in pints, the unit of the other measurement.
No single fact in the customary table links gallons to pints, so go through quarts. Use the fact that 1 gallon equals 4 quarts and the fact that 1 quart equals 2 pints. Each fact pairs two equal, nonzero amounts, so each factor equals 1.
$\frac{4\text{ qt}}{1\text{ gal}} \quad \text{and} \quad \frac{2\text{ pt}}{1\text{ qt}}$
Multiply by both factors. Gallons are on the bottom of the first factor and quarts are on the bottom of the second, so gallons divide out, then quarts divide out, and pints are left. The amount has not changed, because you multiplied by 1 twice.
$2.5\text{ gal} \times \frac{4\text{ qt}}{1\text{ gal}} \times \frac{2\text{ pt}}{1\text{ qt}} = (2.5 \times 4 \times 2)\text{ pt}$
Multiply the numbers from left to right. First, 2.5 times 4 is 10. Then 10 times 2 is 20.
$(2.5 \times 4 \times 2)\text{ pt} = (10 \times 2)\text{ pt} = 20\text{ pt}$
Both measurements are now in pints, so the greater number is the greater amount. Since 20 is less than 22, and 20 pints is the same amount as 2.5 gallons, 2.5 gallons is less than 22 pints.
$20\text{ pt} < 22\text{ pt}, \text{ so } 2.5\text{ gal} < 22\text{ pt}$

Answer

$22$ pints is the greater amount, because $2.5$ gallons equals only $20$ pints.

The metric system is the set of units used in most countries and in science. For length, mass and capacity it starts from three basic units: the meter (m), the gram (g) and the liter (L). A prefix is a word part placed in front of a basic unit to name a larger or smaller unit:

kilo- means one thousand, so $1$ kilometer is $1{,}000$ meters.
centi- means one hundredth, so $100$ centimeters make $1$ meter.
milli- means one thousandth, so $1{,}000$ millimeters make $1$ meter.

The same prefixes work with grams and liters. Every fact in the next table uses $10$ , $100$ or $1{,}000$ , which keeps the arithmetic short.

Metric unit facts: each row gives two names for the same amount

Measures	One larger unit	Equal amount in smaller units
Length	1 kilometer (km)	1,000 meters (m)
Length	1 meter (m)	100 centimeters (cm)
Length	1 meter (m)	1,000 millimeters (mm)
Length	1 centimeter (cm)	10 millimeters (mm)
Mass	1 kilogram (kg)	1,000 grams (g)
Mass	1 gram (g)	1,000 milligrams (mg)
Capacity	1 liter (L)	1,000 milliliters (mL)

Metric conversions use the same method, because each table fact again pairs two equal, nonzero amounts. To rename $0.75$ liters in milliliters, put liters on the bottom so they divide out:

$0.75\text{ L} \times \frac{1{,}000\text{ mL}}{1\text{ L}} = (0.75 \times 1{,}000)\text{ mL} = 750\text{ mL}$

To rename $350$ milligrams in grams, put milligrams on the bottom:

$350\text{ mg} \times \frac{1\text{ g}}{1{,}000\text{ mg}} = \frac{350}{1{,}000}\text{ g} = 0.35\text{ g}$

Check each result by size. A milliliter is smaller than a liter, so the same positive amount takes more milliliters, and $750$ is greater than $0.75$ . A gram is larger than a milligram, so the same positive amount takes fewer grams, and $0.35$ is less than $350$ .

Worked example

Order three metric lengths

Order these lengths from least to greatest: $2.4$ km, $2{,}350$ m and $245{,}000$ cm.

You can order the numbers only after all three measurements use the same unit. Choose meters, because one length is already in meters and the metric table links meters to both other units.
Rename 2.4 kilometers in meters. The fact that 1 kilometer equals 1,000 meters pairs two equal, nonzero lengths, so the factor equals 1. Put kilometers on the bottom so they divide out.
$2.4\text{ km} \times \frac{1{,}000\text{ m}}{1\text{ km}} = (2.4 \times 1{,}000)\text{ m} = 2{,}400\text{ m}$
Rename 245,000 centimeters in meters. Use the fact that 1 meter equals 100 centimeters, with centimeters on the bottom so they divide out. This factor also equals 1.
$245{,}000\text{ cm} \times \frac{1\text{ m}}{100\text{ cm}} = \frac{245{,}000}{100}\text{ m} = 2{,}450\text{ m}$
All three lengths are in meters, so order the numbers. Each has 2 thousands. The number 2,350 has only 3 hundreds, so it is least. Both 2,400 and 2,450 have 4 hundreds, and 0 tens is less than 5 tens.
$2{,}350\text{ m} < 2{,}400\text{ m} < 2{,}450\text{ m}$
Write each length in its original unit, keeping the same order. This is valid because renaming did not change any of the amounts.
$2{,}350\text{ m} < 2.4\text{ km} < 245{,}000\text{ cm}$

Answer

From least to greatest, the lengths are $2{,}350$ m, $2.4$ km and $245{,}000$ cm.

Takeaway: Converting changes the unit that names an amount, not the amount itself. To convert a positive length, weight, mass or capacity, multiply by a conversion factor that equals $1$ , with the unit you are leaving on the bottom so it divides out. To compare measurements, rename them in one shared unit first, then compare the numbers. As a check, the same positive amount always takes fewer of a larger unit and more of a smaller unit.

Next, you will use the same kind of factor to convert between customary and metric units and to solve word problems that involve unit conversion.