Unit Conversion

Unit conversion rarely appears as a question by itself. It appears as the step between a correct calculation and a wrong answer, which makes it the most expensive small skill on the subtest.

Cancel the units

Treat a unit like an algebraic factor. A conversion factor is a fraction equal to 1, and you choose the orientation that cancels what you have.

Convert 4.5 feet to inches.

There are 12 inches in 1 foot, so the factor is either 12 inches over 1 foot, or 1 foot over 12 inches. You have feet, so you need feet on the bottom to cancel them.

4.5 feet times (12 inches / 1 foot). The feet cancel and you are left with 4.5 times 12, which is 54 inches.

That sounds like ceremony for a problem you could do in your head, and it is - until the chain gets to three steps, and then it is the only thing that keeps you straight.

A hose delivers 3 gallons per minute. How many quarts per hour?

Two conversions, and they go in different directions.

3 gallons/minute x (4 quarts / 1 gallon) x (60 minutes / 1 hour)

Gallons cancel against gallons, minutes cancel against minutes, and what survives is quarts over hours: 3 times 4 times 60 is 720 quarts per hour.

Anyone who converts gallons and forgets minutes gets 12, which is a choice. Anyone who converts minutes and forgets gallons gets 180, which is also a choice.

The direction check

Before you compute, decide whether the number should get bigger or smaller.

Going to a smaller unit, the number gets bigger. There are more inches in a length than feet, because an inch is small. So feet to inches multiplies.

Going to a bigger unit, the number gets smaller. Ounces to pounds divides.

This is the check that catches the reversed conversion, which is the most common error in the topic and produces an answer that is off by the square of the factor if it happens twice.

The customary conversions worth memorizing

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These come up. The rest can be reasoned from them.

Length

Weight

Volume

Time

The volume ladder is worth learning as a ladder - gallon, quart, pint, cup - each step being a factor of 2 except the first, which is 4. Going down the ladder multiplies.

Metric is easier, which is the point of it

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Metric prefixes are powers of ten, so conversion is moving a decimal point.

So 1 kilometer is 1,000 meters, 1 meter is 100 centimeters, and 1 centimeter is 10 millimeters.

Convert 2.4 kilometers to centimeters.

Kilometers to meters is times 1,000, giving 2,400 meters. Meters to centimeters is times 100, giving 240,000 centimeters.

Two useful anchors: 1 milliliter of water is 1 cubic centimeter and weighs 1 gram, and 1 liter of water weighs 1 kilogram. Metric was built that way and the test occasionally uses it.

Between systems

The test does not expect precise cross-system conversion, but it expects rough familiarity, and these are the ones that appear.

Only 1 inch to 2.54 centimeters is exact by definition. The rest are rounded, and a question that needs them will give you the factor.

Temperature is the exception

Temperature scales do not share a zero, so they are not a multiplication.

Fahrenheit to Celsius: subtract 32, then multiply by 5/9.

Celsius to Fahrenheit: multiply by 9/5, then add 32.

The order matters both times. Doing the addition first is the standard error.

Two anchors worth holding: water freezes at 32 F and 0 C, and boils at 212 F and 100 C. Minus 40 is the same on both scales, which is a useful oddity for checking that you have the formula the right way round.

What you can skip

Across the 31 questions on this topic:

• Metric conversions. Kilometers never appear and liters and meters once each. The conversions asked are the customary ones: inches, feet, yards, miles, ounces, pounds, cups and gallons.
• Temperature conversion never appears.

Where people lose points

Stopping one conversion short in a chain. The half-converted number is always a choice.

Reversing the factor. Dividing where you should multiply. The direction check above costs three seconds.

Converting a squared or cubed unit with the plain factor. One square foot is 144 square inches, not 12, because both dimensions convert. One cubic foot is 1,728 cubic inches. This appears on area and volume questions and the plain-factor answer is always there.

Doing the temperature steps out of order.

Converting when you did not need to. If every number in the question is in the same unit and the answer choices are too, leave it alone.

Work one in under a minute

A room measures 12 feet by 15 feet. Carpet is sold by the square yard. How many square yards are needed?

Two routes, and one is much better.

The slow route: 12 times 15 is 180 square feet, then divide by 9 - not 3 - because square feet to square yards squares the factor. 180 divided by 9 is 20.

The fast route: convert the sides first. Twelve feet is 4 yards, 15 feet is 5 yards, so the area is 20 square yards.

Converting the dimensions before computing the area sidesteps the squared-factor problem entirely. Do that when you can.

Where this leads

Conversion is the hidden step in nearly every rate and measurement question on the subtest, and the squared-factor rule carries straight into area and volume.