Adding and subtracting fractions

A fraction counts equal pieces

How to solve them

1. 1

   Same denominator: add the tops, keep the bottom

   a/d + b/d = (a + b)/d

   When the denominators are the same, the pieces are the same size. Add (or subtract) the numerators and keep the denominator. The denominator is the name of the piece, like “ninths”. It does not change.

   The most common mistake is adding the bottoms too: 2/9 + 5/9 is not 7/18. Check it with your eyes: 7/18 is less than half, but 2/9 + 5/9 is almost a whole. It cannot be right.

   1. 2/9 + 5/9
   2. (2 + 5)/9
   3. 7/9

   Example

   7/10 − 3/10

   1. Same denominator, 10. Subtract the tops: 7 − 3 = 4.
   2. Keep the bottom: 4/10.

   Answer 4/10, which is 2/5 (step 2)

   Next step: Simplify ↓
2. 2

   Simplify: divide the top and the bottom by the same number

   4/10 and 2/5 are the same amount, cut into different pieces. Writing a fraction with the smallest numbers possible is called simplifying, or writing it in lowest terms.

   Divide the top and the bottom by the same number. Keep going until no number except 1 divides both. If your answer is 6/8 and the choices show 3/4, that is your answer: they are the same number.

   1. 18/24
   2. (18 ÷ 2)/(24 ÷ 2)
   3. 9/12
   4. (9 ÷ 3)/(12 ÷ 3)
   5. 3/4

   Shortcut

   Even? Halve it

   If the top and the bottom are both even, divide both by 2. If both end in 0 or 5, divide both by 5. Then look again: you are done only when nothing but 1 divides both.

   Next step: The LCD ↓
3. 3

   Different denominators: find the least common denominator

   To add 3/4 and 1/6, you first cut both into pieces of the same size. That size is a common denominator: a number that both denominators divide into evenly. The smallest one is the least common denominator, or LCD. Smaller numbers mean easier arithmetic.

   How to find it: list the multiples of the bigger denominator (for 6: 6, 12, 18…) and stop at the first one that the smaller denominator divides into.

   Finding the LCD

   Denominators	Multiples of the bigger one	LCD
   4 and 6	6, 12	12
   8 and 12	12, 24	24
   3 and 5	5, 10, 15	15
   5 and 10	10	10

   Shortcut

   Multiplying always works

   If you can’t find the LCD fast, multiply the two denominators: 4 × 6 = 24 also works for 4 and 6. You will just have bigger numbers to simplify at the end.

   Next step: Rewrite ↓
4. 4

   Rewrite each fraction, then add or subtract

   To change a denominator without changing the amount, multiply the top and the bottom by the same number. To write 3/4 with a 12 on the bottom: 4 × 3 = 12, so the top is multiplied by 3 too, and 3/4 becomes 9/12.

   Now the pieces are the same size, and step 1 does the rest. Simplify at the end.

   1. 3/4 + 1/6
   2. 9/12 + 1/6
   3. 9/12 + 2/12
   4. 11/12

   Example

   7/8 − 5/12

   1. The LCD: the multiples of 12 are 12, 24. 24 ÷ 8 = 3, so the LCD is 24.
   2. 7/8 = 21/24 (top and bottom × 3). 5/12 = 10/24 (top and bottom × 2).
   3. Subtract the tops: 21 − 10 = 11, so 11/24. Nothing but 1 divides both 11 and 24, so it is already simplified.

   Answer 11/24

   Next step: Mixed numbers ↓
5. 5

   Mixed numbers: wholes with wholes, pieces with pieces

   A mixed number is a whole number and a fraction together: 2 3/5 means 2 wholes and 3/5 more. To add two of them, add the whole numbers, add the fractions, and put them together.

   A fraction with a top bigger than its bottom, like 7/5, is called an improper fraction: it is more than one whole. To turn it into a mixed number, ask how many times the bottom goes into the top. 5 goes into 7 once, with 2 left over, so 7/5 = 1 2/5. If the fractions add up to more than one whole, carry that whole over to the whole numbers.

   Going the other way: whole × bottom + top, over the same bottom. 2 3/5 is 2 × 5 + 3 = 13 fifths, or 13/5.

   1. 2 3/5 + 1 4/5
   2. 3 + 3/5 + 4/5
   3. 3 + 7/5
   4. 3 + 1 2/5
   5. 4 2/5

   Example

   1 3/4 + 2 5/6

   1. Wholes: 1 + 2 = 3.
   2. Fractions, with the LCD 12: 3/4 = 9/12 and 5/6 = 10/12, so 9/12 + 10/12 = 19/12.
   3. 19/12 is 1 whole and 7/12. Carry the 1: 3 + 1 7/12 = 4 7/12.

   Answer 4 7/12

   Next step: Borrowing ↓
6. 6

   Subtracting mixed numbers: borrow a whole when you need to

   Subtract wholes from wholes and fractions from fractions. The problem: sometimes the first fraction is smaller than the one you take away. In 5 1/4 − 2 3/4 you cannot take 3/4 from 1/4.

   Borrow (also called regrouping): take 1 from the whole number and cut it into pieces. 1 whole = 4/4, so 5 1/4 becomes 4 + 4/4 + 1/4, which is 4 5/4. Same amount, written another way. Now you can subtract.

   1. 5 1/4 − 2 3/4
   2. 4 5/4 − 2 3/4
   3. (4 − 2) + (5/4 − 3/4)
   4. 2 + 2/4
   5. 2 1/2

   Example

   6 − 2 3/8

   1. 6 has no fraction to take 3/8 from. Borrow 1: 6 = 5 8/8.
   2. Wholes: 5 − 2 = 3. Fractions: 8/8 − 3/8 = 5/8.

   Answer 3 5/8

   Next step: Three examples ↓
7. 7

   Three worked examples, from easy to test level

   Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

   In word problems, “in all”, “total” and “combined” mean add. “How much more”, “how much is left” and “difference” mean subtract. The test-level example asks for both, one after the other: write down the first result before you start the second.

   Example · Easy

   What is 3/10 + 1/2?

   First try3/10 + 1/2 = 4/12

   1. 4/12 is one third, but 1/2 alone is already bigger than that. Adding the bottoms cannot be right: the pieces have to be the same size first.
   2. The LCD: 10 is a multiple of 2, so the LCD is 10. 1/2 = 5/10 (top and bottom × 5).
   3. Add the tops: 3/10 + 5/10 = 8/10.
   4. Simplify, dividing top and bottom by 2: 8/10 = 4/5.

   Answer 4/5

   Example · Medium

   A cook has 3 1/4 cups of rice. A recipe uses 1 2/3 cups. How much rice is left?

   First try3 3/12 − 1 8/12 → 8/12 − 3/12 = 5/12, so 2 5/12

   1. The fractions were flipped. The subtraction is 3/12 − 8/12, and you cannot take 8 twelfths from 3 twelfths. Flipping them gives a wrong answer that looks fine.
   2. The LCD of 4 and 3 is 12: 3 1/4 = 3 3/12 and 1 2/3 = 1 8/12.
   3. Borrow 1 whole = 12/12: 3 3/12 = 2 15/12.
   4. Wholes: 2 − 1 = 1. Fractions: 15/12 − 8/12 = 7/12. Rice left: 1 7/12 cups.

   Answer 1 7/12 cups

   Example · Test level

   A squad hikes 2 3/8 miles before lunch and 1 3/4 miles after lunch. The whole route is 6 miles. How many miles are left?

   First try2 3/8 + 1 3/4 = 3 6/12 = 3 1/2, so 6 − 3 1/2 = 2 1/2

   1. 3/8 + 3/4 is not 6/12: the bottoms are never added. Eighths and fourths have to become the same size first.
   2. Hiked so far: 3/4 = 6/8, so 2 3/8 + 1 6/8 = 3 9/8. 9/8 is 1 1/8, so carry the 1: 4 1/8 miles.
   3. Left: 6 − 4 1/8. Borrow 1 from the 6: 6 = 5 8/8.
   4. Wholes: 5 − 4 = 1. Fractions: 8/8 − 1/8 = 7/8. Miles left: 1 7/8.

   Answer 1 7/8 miles

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.