1 · Equivalent fractions and simplest form
Multiplying or dividing the top and bottom by the same number gives the same amount: 2/3 = 4/6 = 8/12.
A fraction is in simplest form when the top and bottom share no factor except 1. 18/24: both divide by 6, so 18/24 = 3/4.
2 · Comparing fractions
Same denominator: compare the tops. 5/8 > 3/8. Same numerator: the smaller denominator is bigger, because the pieces are bigger: 3/5 > 3/7.
Otherwise, rewrite with a common denominator. 5/6 or 7/9? Use 18: 15/18 and 14/18, so 5/6 is greater. Another check: compare each to 1/2. 4/9 is less than 1/2 (4.5/9), and 5/8 is more (4/8).
3 · Mixed numbers and improper fractions
2 3/4 means 2 wholes and 3/4. Each whole is 4 fourths, so 2 3/4 = 8/4 + 3/4 = 11/4.
Going back: 17/5 is 17 ÷ 5 = 3 R2, so 17/5 = 3 2/5.
4 · Adding with unlike denominators
2/3 + 1/4. Thirds and fourths are different sized pieces, so change both to twelfths: 2/3 = 8/12 and 1/4 = 3/12. Then 8/12 + 3/12 = 11/12.
Never add the tops and the bottoms. 2/3 + 1/4 is not 3/7; 3/7 is less than 2/3, and adding should make it bigger.
5 · Subtracting with regrouping
5 1/4 − 2 3/4. You cannot take 3/4 from 1/4, so borrow one whole: 5 1/4 = 4 5/4.
Now 4 5/4 − 2 3/4 = 2 2/4 = 2 1/2. Subtracting "the small fraction from the big one" (3/4 − 1/4) gives 3 2/4, which is wrong.
6 · A fraction of a number
3/5 of 40: find 1/5 first (40 ÷ 5 = 8), then take 3 of them: 3 × 8 = 24.
Divide by the bottom, multiply by the top. It works for money and measures too: 3/4 of $20 is $15.
7 · "Of what is left"
"Leo had 60 stickers. He gave 1/3 to his sister, then 1/4 of the rest to a friend." First: 1/3 of 60 = 20, leaving 40. Then 1/4 of 40 = 10, leaving 30.
The trap is taking 1/4 of the 60. Read whose amount each fraction is "of".
8 · Working backwards from a fraction
"2/5 of a class is 12 students. How many are in the class?" 2 fifths = 12, so 1 fifth = 6, and 5 fifths = 30.
Check: 2/5 of 30 = 12.
9 · Multiplying fractions
Multiply tops and multiply bottoms: 2/3 × 3/4 = 6/12 = 1/2.
Multiplying by a fraction less than 1 makes a number smaller. "Half of 3/4" is 1/2 × 3/4 = 3/8.
10 · Dividing by a fraction
"How many 3/4-cup scoops are in 6 cups?" That is 6 ÷ 3/4. Each cup holds 4 fourths, so 6 cups = 24 fourths, and 24 fourths ÷ 3 fourths = 8 scoops.
The short way: flip the fraction and multiply. 6 ÷ 3/4 = 6 × 4/3 = 24/3 = 8. Dividing by a number less than 1 makes the answer bigger.
11 · Fractions on the ISEE
Look for the fraction that is easiest to use. Many questions are about halves, thirds, fourths, and fifths of numbers that divide evenly.
Before you choose, ask whether your answer should be bigger or smaller than the numbers in the question. That one check catches most trap answers.
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