Fractions of a Quantity

Percent questions and fraction questions are the same question wearing different clothes, with one difference that matters for the answer: percent questions usually want a percent back, and fraction questions usually want a quantity - dollars, people, gallons, hours. That changes what a correct answer looks like and it changes which distractors are waiting.

Taking one fraction

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A fraction of an amount is a multiplication. Nothing else is happening.

A squadron of 96 airmen is at the range. Three eighths qualify expert. How many is that?

Three eighths of 96 is 3/8 times 96.

Do not multiply 3 by 96 and then divide by 8. Divide first: 96 divided by 8 is 12, then 12 times 3 is 36. Same answer, smaller numbers, and no long division under time pressure.

That is the general move. Divide by the denominator, multiply by the numerator. The denominator tells you how big one piece is; the numerator tells you how many pieces you want.

If the denominator does not divide the amount evenly, multiply first and divide at the end, and keep the fraction rather than converting to a decimal until you have to.

Chaining two fractions

This is the shape the subtest actually asks, and it is worth slowing down for.

A recruiter has 120 leads. She reaches one third of them on the first call. Of those she did not reach, half answer on the second call. How many did she reach on the second call?

The trap is the phrase "of those she did not reach". The second fraction is not half of 120. It is half of what was left.

Step by step:

• Reached first: 1/3 of 120 is 40.
• Not reached: 120 minus 40 is 80.
• Reached second: half of 80 is 40.

The answer built on the original amount - half of 120, or 60 - will be sitting in the choices, and it is the most common wrong answer on this question type.

The one-step version

Once you see the structure, you can skip the middle.

If one third was reached, then two thirds were not. Half of two thirds is 2/3 times 1/2, which is 2/6, or one third. One third of 120 is 40.

That is one multiplication instead of three operations. The move is to convert "what is left" into a fraction immediately:

Fraction remaining = 1 minus the fraction taken.

Then multiply the remaining fraction by the next fraction and apply the result to the original amount, once.

Three stages, still one multiplication

The method does not get harder when the problem does.

A unit starts a 240-mile move. It covers 1/4 of the distance on the first day. On the second day it covers 2/3 of what remains. How far is left after two days?

Remaining after day one: 1 minus 1/4 is 3/4.

Day two covers 2/3 of that, so what is left after day two is 1/3 of the 3/4.

One third of three quarters is 1/3 times 3/4, which is 3/12, or 1/4.

One quarter of 240 is 60 miles.

Notice the question asked what is left, not what was covered. Had it asked how far the unit traveled on day two, the answer would be 2/3 of 3/4 of 240, which is 120. Both numbers are in the choices. Re-read the question.

Fractions and mixed numbers in the setup

Word problems hand you mixed numbers because that is how people talk. Convert before you multiply.

Concrete is poured at 2 1/4 cubic yards per hour for 5 1/3 hours. How much was poured?

Two and a quarter is 9/4. Five and a third is 16/3.

9/4 times 16/3: cancel the 3 into the 9 to get 3, and cancel the 4 into the 16 to get 4. That leaves 3 times 4 over 1, which is 12 cubic yards.

Cancelling first turned a four-digit multiplication into 3 times 4. On a test with no calculator that is most of the work.

What you can skip

Across the 7 questions on this topic:

• Complex fractions. The seven questions on this topic take a simple fraction of a whole or find the whole from a part; nothing more elaborate.

Where people lose points

Applying the second fraction to the original. The big one. "Of the remainder", "of those left", "of the rest" all mean the base has changed.

Answering the part that was taken when the question asked for what is left, or the reverse. Both numbers are choices.

Adding fractions when the problem chains them. "One third, then half of what is left" is not 1/3 plus 1/2. Chained fractions multiply.

Converting to decimals too early. One third is 0.333..., and rounding it partway through a three-step problem will land you between two answer choices. Keep the fraction.

Multiplying before cancelling. Not wrong, just slow, and slow is how arithmetic errors happen.

Work one in under a minute

A 3,600-gallon tank is 5/6 full. Two fifths of the fuel is issued. How many gallons remain in the tank?

Currently in the tank: 5/6 of 3,600. One sixth is 600, so five sixths is 3,000.

Two fifths issued means three fifths remain. Three fifths of 3,000: one fifth is 600, so three fifths is 1,800 gallons.

Check: it should be less than 3,000 and more than half of it. It is.

Where this leads

The remaining-fraction move here is the same move as working with a base in percent problems, and the same move as scaling a ratio to a known total.