Measurement and Unit Conversion

Measurement and Unit Conversion

Unit conversion questions reward careful, organized work. The arithmetic is simple, but one dropped factor produces an answer that looks reasonable and is wrong, and the answer choices are built to catch exactly that. This chapter shows how to convert length, area, volume, and compound rates without losing a factor.

Linear conversions

To convert a length, multiply or divide by the conversion factor once. Ask whether the new unit is bigger or smaller than the old one. If you are converting to a smaller unit (yards to feet), you expect a larger number, so multiply. If you are converting to a larger unit (grams to kilograms), you expect a smaller number, so divide.

• $8$ yards $=8\times 3=24$ feet.
• $9$ feet $=9\times 12=108$ inches.
• $7{,}500$ grams $=7{,}500\div 1{,}000=7.5$ kilograms.
• $2.5$ hours $=2.5\times 60=150$ minutes.

It helps to memorize the standard facts: $1$ mile $=5{,}280$ feet $=1{,}760$ yards, $1$ yard $=3$ feet $=36$ inches, $1$ foot $=12$ inches, $1$ hour $=60$ minutes $=3{,}600$ seconds, and $1$ kilogram $=1{,}000$ grams. The ACT usually supplies unusual conversions in the problem, but it expects you to know the common ones.

Squared and cubed units

This is where most errors happen. Area is measured in square units, so converting an area means applying the conversion factor in both dimensions. Volume is measured in cubic units, so the factor applies three times.

• Converting an area: square the linear factor.
• Converting a volume: cube the linear factor.

Worked example. Convert 3 square feet to square inches.

One foot is $12$ inches, so one square foot is $12^2=144$ square inches. Then $3\times 144=432$ square inches.

The picture to keep in mind is a square that is 1 foot on a side. It is also 12 inches on a side, so it contains $12\times 12=144$ one-inch squares. In the same way, a cubic yard is $3\times 3\times 3=27$ cubic feet, and $4$ cubic yards is $4\times 27=108$ cubic feet. Using only the linear factor ($3$ here) is the classic wrong answer.

Compound rates

A compound rate such as miles per hour has two units, and converting to feet per second changes both. Handle them separately and then combine: convert the distance unit on top, convert the time unit on the bottom.

Worked example. Convert 30 miles per hour to feet per second.

Miles to feet means multiplying by $5{,}280$. Hours to seconds means the rate is spread over $3{,}600$ seconds, so divide by $3{,}600$:

$30\times 5{,}280\div 3{,}600=44$ feet per second.

A handy check: $60$ miles per hour is exactly $88$ feet per second, so $30$ miles per hour should be half of that, which is $44$. The same factors work backward: $88\times 3{,}600\div 5{,}280=60$ miles per hour, since multiplying by the seconds and dividing by the feet goes the other way.

Conversions inside real problems

Many questions ask you to apply a rate to a real space. A paint that covers $350$ square feet per gallon needs $\frac{1{,}400}{350}=4$ gallons to cover a $1{,}400$-square-foot room: divide the total area by the coverage rate. If the room were measured in different units than the coverage, you would convert first, then divide. Write the unit next to every number so you can see whether the units cancel to leave the one you want.

The unit-cancelling method

Write the quantity as a fraction and multiply by conversion fractions that equal $1$, arranged so unwanted units cancel. To convert $6$ yards to inches:

$6\text{ yd}\times\frac{36\text{ in}}{1\text{ yd}}=216\text{ in}$.

The yards cancel and inches remain. If your unit does not cancel, flip the fraction. This method also works for compound rates, and for squared units you simply include the fraction twice.

Common traps, side by side

Consider converting $4$ cubic feet to cubic inches. Using the linear factor gives $4\times 12=48$, which is far too small: a cubic inch is tiny compared with a cubic foot, so the count should be large. Cubing the factor gives $12^3=1{,}728$ cubic inches per cubic foot, and $4\times 1{,}728=6{,}912$. In the same way, $2$ square yards in square inches is $2\times 36^2=2\times 1{,}296=2{,}592$, not $2\times 36=72$. When a conversion factor is not given, build it from smaller facts: $1$ yard $=3$ feet $=36$ inches. Direction errors are the second trap. Converting $9$ feet to inches by dividing by $12$ gives $0.75$, which is smaller than the original number even though inches are the smaller unit. Checking whether the answer should grow or shrink catches this immediately.

Strategy

• Before calculating, predict whether the number should grow or shrink.
• For area, square the factor; for volume, cube it.
• For a compound rate, convert each unit separately.
• Write units on every line of work.
• Check the final answer for reasonableness against a benchmark you know, such as $60$ mph $=88$ ft/s.
• Reread the question: the problem often gives a quantity in one unit and asks for another, so the final conversion is a required step.