Fractions, Decimals and Percents

Three notations for one idea. The test moves between them freely and expects you to as well, usually because one form makes a calculation trivial and the other two make it long.

The conversions worth knowing on sight

Memorize this table. It pays back constantly.

The eighths are the ones people skip and the ones the test likes, because they produce decimals that look awkward and are not.

Converting anything else

Fraction to decimal: divide the top by the bottom.

Decimal to percent: move the point two places right. 0.45 becomes 45 percent.

Percent to decimal: move it two places left. 8 percent becomes 0.08.

Decimal to fraction: the digits go on top, and the place value of the last digit goes on the bottom. 0.36 is 36/100, which reduces to 9/25.

Percent to fraction: put it over 100 and reduce. 40 percent is 40/100, which is 2/5.

Adding and subtracting

You need a common denominator. There is no way around it.

3/8 plus 1/6

Least common multiple of 8 and 6 is 24.

• 3/8 becomes 9/24.
• 1/6 becomes 4/24.
• Sum: 13/24, which does not reduce.

Multiply each fraction top and bottom by whatever turns its denominator into the LCM. Multiplying top and bottom by the same thing does not change the value, which is why this is legal.

Adding the tops and the bottoms separately is the error. 3/8 plus 1/6 is not 4/14. That operation has a name in statistics and it is not addition.

Mixed numbers

Convert to improper fractions, operate, convert back.

3 1/4 minus 1 5/6

Three and a quarter is 13/4; one and five sixths is 11/6. Over 12: 39/12 minus 22/12 is 17/12, which is 1 5/12.

You can also subtract the whole parts and the fractions separately, but that needs borrowing when the second fraction is larger, and borrowing a 1 as 12/12 is where people go wrong. Converting first is slower to write and harder to get wrong.

Multiplying

No common denominator needed. Tops times tops, bottoms times bottoms.

3/8 times 4/9

Before multiplying, cancel: the 4 and the 8 share a 4, leaving 1 and 2; the 3 and the 9 share a 3, leaving 1 and 3.

That gives 1/2 times 1/3, which is 1/6.

Multiplying first gives 12/72, which reduces to the same thing after a division you did not need to do. Cancel diagonally across the multiplication sign - any top with any bottom.

Dividing

Flip the second fraction and multiply.

3/5 divided by 9/10

Becomes 3/5 times 10/9. Cancel the 5 into the 10 to get 2, and the 3 into the 9 to get 3: 1/1 times 2/3 is 2/3.

Only the second fraction flips. Flipping both, or flipping the first, are the two standard errors, and both produce answers that are in the choices.

A whole number divided by a fraction is the same move: 6 divided by 2/3 is 6 times 3/2, which is 9. Dividing by a fraction less than 1 makes the number bigger, which is the sanity check that catches a flip in the wrong place.

Comparing fractions

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Three ways, fastest first.

Same denominator? Compare the tops.

Same numerator? The bigger denominator is the smaller fraction. 3/7 is less than 3/5, because sevenths are smaller pieces.

Neither? Cross-multiply upward. To compare 5/8 and 7/11, multiply 5 by 11 to get 55 and 7 by 8 to get 56. The 56 came from the 7/11 side, so 7/11 is larger.

Converting both to decimals also works and is often easier if the conversions are in the table above.

Percent change

About a dozen questions here are percent change. Change divided by the original, times 100.

What percent increase is 50 to 60? The change is 10, the original 50: 20 percent.

80 to 60 is a change of 20 from 80: a 25 percent decrease.

The trap is the base. Always divide by the starting value. And two changes do not simply add: up 25 percent and then down 20 percent is 1.25 x 0.80 = 1.00 - no change at all. The Arithmetic Reasoning lesson on percent change works through more of these.

What you can skip

Across the 187 questions on this topic:

• Repeating decimal conversions never appear.
• Long chains of mixed numbers. Mixed numbers come up three times; convert to improper fractions and the rules above apply.

Where people lose points

Adding denominators. The most common fraction error there is.

Finding a common denominator when multiplying. Not wrong, just wasted time, and the extra arithmetic invites errors.

Flipping the wrong fraction on division.

Not reducing the final answer. Choices are in lowest terms, and an unreduced correct answer looks like a wrong answer.

Losing the whole number when converting a mixed number, particularly on subtraction.

Moving the decimal the wrong way on a percent conversion. Percent to decimal makes the number smaller.

Work one in under a minute

Evaluate 2/3 plus 1/4, then divide the result by 11/6.

Sum: over 12, that is 8/12 plus 3/12, which is 11/12.

Divide: 11/12 divided by 11/6 is 11/12 times 6/11. The 11s cancel and the 6 goes into the 12 twice, leaving 1/2.

Check: dividing by 11/6, which is bigger than 1, should shrink the 11/12. It did.

Where this leads

Fraction fluency decides how fast you move through every other topic in this subtest, and it is the difference between finishing the section and not.