Use Ratios to Convert Measurement Units | Grade 6 Math | LevelKite

Picture a ruler with inches on one edge and centimeters on the other. Both edges measure the same pencil. The length stays the same. Only the unit and the number change.

A conversion fact works the same way. You already know facts such as $1 \text{ ft} = 12 \text{ in}$ . Each side names the same length. In this lesson you will treat a fact like this as a ratio and use it to convert units, even between two measurement systems.

A ratio compares two quantities. It tells how much of one quantity goes with a given amount of the other. The fact $1 \text{ ft} = 12 \text{ in}$ gives the ratio $1$ foot to $12$ inches, written $1 \text{ ft} : 12 \text{ in}$ .

A ratio table lists pairs of numbers that share one ratio. Read each row below as one length named in two units.

One length named in feet and in inches

Feet	Inches	How the row is made from row 1
$1$	$12$	the conversion fact
$2$	$24$	both numbers $\times 2$
$5$	$60$	both numbers $\times 5$
$0.5$	$6$	both numbers $\div 2$
$4.5$	$54$	both numbers $\times 4.5$

The rows show equivalent ratios: ratios that describe the same relationship. You make one by multiplying or dividing both numbers of a ratio by the same number greater than zero. This is called scaling the ratio.

Why does each row still name one length? Every foot is made of $12$ inches. So $5$ feet are $5$ groups of $12$ inches, and half a foot is half of $12$ inches. Scaling keeps every foot matched with its $12$ inches.

Now connect the table to symbols. Let $f$ be the number of feet in a length. Let $i$ be the number of inches in the same length. In every row, $i$ is $12$ times $f$ . The number $12$ is the unit rate: the number of inches for each $1$ foot.

Multiplying by $12$ and dividing by $12$ are inverse operations: each one undoes the other. This works because $12$ is not zero. So the two equations below describe the same relationship, read in two directions.

i = 12 \times f \qquad \text{and} \qquad f = i \div 12

Key concept

A conversion fact is a ratio you can scale

For the units in this lesson, which measure length, weight or capacity (how much a container holds), a conversion fact pairs two measures of the same amount. Scale the ratio: multiply or divide both of its numbers by the same number greater than zero. You get an equivalent ratio, so it names the same amount in two units.

In the formula, unit A and unit B are the two units in the fact, and $a$ and $b$ are its two numbers. The scale factor $k$ is the number you multiply both by. It is greater than zero. To find $k$ , you divide by a number from the fact, and that divisor is never zero.

For any amount greater than zero, it takes more of the smaller unit than of the larger unit.

a \text{ of unit A} = b \text{ of unit B} \quad\Rightarrow\quad (a \times k) \text{ of unit A} = (b \times k) \text{ of unit B}

Check every division. In $54 \div 12 = 4.5$ , the dividend is $54$ , the divisor is $12$ and the quotient is $4.5$ . Every divisor in a conversion is greater than zero, so you never divide by zero.

Exact quotient: quotient $\times$ divisor equals the dividend: $4.5 \times 12 = 54$ .
Rounded quotient: quotient $\times$ divisor only comes close to the dividend. The check is approximate.
Split or not: a length can be split, so $4.5$ ft is a fractional amount with no remainder. Use a remainder, the amount left over, only when you count whole units and leave the rest unsplit: $4$ whole feet with $6$ in left over. Check that with $12 \times 4 + 6 = 54$ .
Comes out even: $48 \div 12 = 4$ exactly, so there is no remainder to write.

Worked example

Quarts to gallons

Convert $15$ quarts to gallons. Use the fact $4$ quarts $= 1$ gallon.

Write the fact as a ratio, because it pairs two measures of the same amount of liquid. Let g be the number of gallons in 15 quarts. You need an equivalent ratio that has 15 quarts.
$4 \text{ qt} : 1 \text{ gal} = 15 \text{ qt} : g \text{ gal}$
Let k be the scale factor that turns 4 quarts into 15 quarts, so 4 × k = 15. Division undoes multiplication, so divide 15 by 4. The divisor 4 is not zero. To divide, use 4 × 3 = 12, which leaves 15 - 12 = 3. Then 3 ÷ 4 = 0.75, so the quotient is 3 + 0.75 = 3.75.
$k = 15 \div 4 = 3.75$
Multiply both numbers of the ratio by 3.75. Scaling both by the same number greater than zero gives an equivalent ratio, so the amount of liquid does not change.
$(4 \times 3.75) \text{ qt} : (1 \times 3.75) \text{ gal} = 15 \text{ qt} : 3.75 \text{ gal}$
Check the division. The quotient 3.75 is exact, so quotient × divisor must equal the dividend: 3 × 4 = 12, 0.75 × 4 = 3, and 12 + 3 = 15. A gallon can be split, so the 3 leftover quarts became 0.75 of a gallon, a fractional part and not a remainder. A gallon is larger than a quart, so needing fewer gallons than quarts makes sense.
$3.75 \times 4 = 15$

Answer

$15$ quarts equals $3.75$ gallons.

So far both units came from one system. The customary system, used in the United States, has units such as inches, miles, pounds and gallons. The metric system has units such as centimeters, kilometers, kilograms and liters.

For length, weight and capacity, a conversion fact can also link a customary unit to a metric unit. It is still a ratio of two measures of one amount, so you scale it the same way.

One thing is new. Most facts in the table below are rounded. The symbol $\approx$ means is approximately equal to. When you use a rounded fact, treat your answer as approximate: write $\approx$ or say about.

Conversion facts that link customary and metric units

What is measured	Conversion fact	Exact or rounded?
Length	$1 \text{ in} = 2.54 \text{ cm}$	exact
Length	$1 \text{ mi} \approx 1.61 \text{ km}$	rounded
Weight or mass	$1 \text{ lb} \approx 0.454 \text{ kg}$	rounded
Capacity	$1 \text{ qt} \approx 0.946 \text{ L}$	rounded
Capacity	$1 \text{ gal} \approx 3.79 \text{ L}$	rounded

Worked example

Pounds to kilograms

A bag of flour is labeled $5$ pounds. About how many kilograms is that? Use $1 \text{ lb} \approx 0.454 \text{ kg}$ .

Write the fact as a ratio. Let m be the number of kilograms in 5 pounds. You need an equivalent ratio that has 5 pounds. The fact is rounded, so the result will be approximate.
$1 \text{ lb} : 0.454 \text{ kg} = 5 \text{ lb} : m \text{ kg}$
Let k be the scale factor that turns 1 pound into 5 pounds, so 1 × k = 5. Then k is 5 ÷ 1 = 5. The divisor 1 is not zero, and the quotient is exact because 5 × 1 = 5.
$k = 5 \div 1 = 5$
Multiply both numbers of the ratio by 5, so the new pair still names the same amount. Multiply 0.454 in parts: 5 × 0.4 = 2, 5 × 0.05 = 0.25 and 5 × 0.004 = 0.02. Add the parts: 2 + 0.25 + 0.02 = 2.27.
$(1 \times 5) \text{ lb} : (0.454 \times 5) \text{ kg} = 5 \text{ lb} : 2.27 \text{ kg}$
Decide if the answer makes sense. A kilogram is a larger unit than a pound, so you need fewer kilograms than pounds, and 2.27 is less than 5. The fact was rounded, so write the result with the approximately-equal symbol.
$5 \text{ lb} \approx 2.27 \text{ kg}$

Answer

A $5$ -pound bag of flour is about $2.27$ kilograms.

Worked example

Centimeters to inches

A ruler is $30$ centimeters long. About how many inches long is it? Use $1 \text{ in} = 2.54 \text{ cm}$ , and round to the nearest tenth of an inch.

Write the fact as a ratio. Let n be the number of inches in 30 centimeters. You need an equivalent ratio that has 30 centimeters.
$1 \text{ in} : 2.54 \text{ cm} = n \text{ in} : 30 \text{ cm}$
Let k be the scale factor that turns 2.54 centimeters into 30 centimeters, so 2.54 × k = 30. Division undoes multiplication, so divide 30 by 2.54. The divisor 2.54 is not zero.
$k = 30 \div 2.54$
Divide. Multiply both numbers by 100. Multiplying both by the same nonzero number keeps the quotient the same: 30 ÷ 2.54 = 3000 ÷ 254. Now 254 × 11 = 2794 and 3000 - 2794 = 206. For tenths, 2060 ÷ 254 gives 8, since 254 × 8 = 2032, with 28 left. For hundredths, 280 ÷ 254 gives 1. So k begins 11.81, which rounds to 11.8.
$k = 3000 \div 254 \approx 11.8$
Multiply both numbers of the ratio by k. The inch number in the fact is 1, so n is 1 × k.
$n = 1 \times k \approx 11.8$
Check the division. The quotient was rounded, so quotient × divisor should come close to the dividend 30, not equal it: 10 × 2.54 = 25.4, 1.8 × 2.54 = 4.572, and 25.4 + 4.572 = 29.972. That is close to 30. An inch is longer than a centimeter, so needing fewer inches than centimeters also makes sense.
$11.8 \times 2.54 = 29.972 \approx 30$

Answer

A $30$ -centimeter ruler is about $11.8$ inches long.

Takeaway. For length, weight and capacity, a conversion fact is a ratio: two measures of one amount. To convert, scale both numbers by the same number greater than zero. You find the scale factor by dividing by a number in the fact, which is never zero. This works inside one system and between the customary and metric systems.

Check each division. An exact quotient times the divisor equals the dividend, and a rounded quotient only comes close. A measurement can be split, so a leftover becomes a fraction or decimal of the unit. A remainder is only for whole units with the rest left unsplit, checked with divisor $\times$ whole quotient $+$ remainder.

Next, you will solve conversion word problems, track units as you multiply and divide, and spot mistakes in unit conversions.