Fractions & Decimals

1 · Common denominators

To add or subtract fractions, the pieces must be the same size. 5/6 − 3/8: the least common denominator of 6 and 8 is 24, so 20/24 − 9/24 = 11/24.

Never add or subtract the tops and the bottoms separately. 1/2 + 1/3 is 5/6, not 2/5.

2 · Mixed numbers

6 1/4 − 2 5/8: rewrite with eighths, 6 2/8 − 2 5/8. You cannot take 5/8 from 2/8, so borrow a whole: 5 10/8 − 2 5/8 = 3 5/8.

For multiplying or dividing, change mixed numbers to improper fractions first. 2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 3 1/3, not 2 × 1 plus 1/2 × 1/3.

3 · Multiplying and dividing

Multiply straight across: 3/4 × 2/9 = 6/36 = 1/6. Simplify before you multiply when you can; it keeps the numbers small.

To divide, multiply by the reciprocal (the flip) of the second fraction: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 1 1/4. Dividing by a number less than 1 makes the answer bigger.

4 · Fractions with signs

The sign rules for integers work for fractions too. −2/3 × 3/5 = −6/15 = −2/5, and −1/2 ÷ (−1/4) = −1/2 × (−4) = 2.

−3/4 is the same as −(3/4) and 3/(−4). The negative sign belongs to the whole fraction.

5 · Fractions, decimals, and percents

A fraction is a division: 7/8 = 7 ÷ 8 = 0.875 = 87.5%. Know the common ones by heart: 1/8 = 0.125, 1/5 = 0.2, 1/4 = 0.25, 1/3 = 0.333…, 2/3 = 0.666…

Some fractions make repeating decimals. 5/11 = 0.4545…, because the division never ends. Writing the top and bottom side by side (0.511) is not division.

6 · Ordering rational numbers

Change everything to decimals, then place them on a number line. −0.7, −2/3, −5/8: as decimals, −0.7, −0.667, −0.625. The one farthest left is −0.7, so it is the least.

For negatives, the number with the bigger size is the smaller number. That is the opposite of what happens with positives.

7 · Finding a fraction between two others

Which fraction is between 3/5 and 2/3? As decimals they are 0.6 and 0.667, so look for a choice in that gap. 5/8 = 0.625 works.

Another way: use a common denominator. 3/5 = 18/30 and 2/3 = 20/30, so 19/30 is between them.

8 · "Of what is left"

Sam spent 1/4 of his money on a game and then 1/3 of what was left on lunch. First: 1/4 is gone, so 3/4 is left. Then 1/3 of that 3/4 is 1/4. He spent 1/4 + 1/4 = 1/2 in all, so 1/2 is left.

The trap is taking 1/3 of the whole amount. Ask "a fraction of what?" each time.

9 · Working backwards

A tank is 3/8 full and holds 27 gallons. Then 3/8 of the tank is 27 gallons, so 1/8 is 9 gallons and the whole tank holds 8 × 9 = 72 gallons.

If 2/5 of a class walks, 1/3 rides the bus, and the other 8 students get a ride, then 2/5 + 1/3 = 11/15, so the 8 students are 4/15 of the class. 1/15 is 2 students and the class has 30.

10 · On the ISEE

Estimate before you compute. 7/8 × 5/6 must be a little less than 1, so an answer like 35/14 cannot be right. A quick size check rules out most trap choices.