Multiplying and dividing fractions

Multiplying by a fraction means taking a part

How to solve them

1. 1

   Multiply straight across: top times top, bottom times bottom

   a/b × c/d = (a × c) / (b × d)

   Adding fractions needs a common denominator. Multiplying does not. Multiply the two tops to get the new top. Multiply the two bottoms to get the new bottom. Then simplify if you can.

   A whole number is a fraction with 1 on the bottom: 4 is the same as 4/1. That lets you multiply a fraction by a whole number in the same way.

   1. 2/3 × 4/5
   2. (2 × 4) / (3 × 5)
   3. 8/15

   Example

   4 × 2/7

   1. Write 4 as a fraction: 4/1.
   2. 4/1 × 2/7 = (4 × 2)/(1 × 7) = 8/7.
   3. 8/7 is more than 1: 7/7 is one whole, and 1/7 is left over. That is 1 1/7 (one and one seventh).

   Answer 8/7 = 1 1/7

   Next step: Cancel first ↓
2. 2

   Simplify before you multiply: cancel

   Big tops and big bottoms make big products that are hard to simplify. So simplify before you multiply. If a top number and a bottom number can both be divided by the same number, divide them both. This is called canceling.

   It works across the two fractions too: the top of one with the bottom of the other. In 9/14 × 7/12, the 9 (top) and the 12 (bottom) both divide by 3. The 7 (top) and the 14 (bottom) both divide by 7.

   Key idea

   A top with a bottom, never a top with a top

   Canceling is the same as simplifying at the end, done early, so it must always pair a numerator with a denominator. Two tops, or two bottoms, never cancel.

   1. 9/14 × 7/12
   2. 3/14 × 7/4
   3. 3/2 × 1/4
   4. 3/8

   Next step: Part of a number ↓
3. 3

   A fraction of a number: divide by the bottom, multiply by the top

   “3/8 of 56” means 3/8 × 56. You could write 56 as 56/1 and multiply across, but there is a faster way that keeps the numbers small: first find the size of one part, then take as many parts as the top says.

   ÷ bottom

   Cut the number into equal parts.

   56 ÷ 8 = 7

   × top

   Take that many parts.

   7 × 3 = 21

   Check

   A part is less than the whole.

   21 is less than 56

   1. 3/8 of 56
   2. 3 × (56 ÷ 8)
   3. 3 × 7
   4. 21

   Example

   A $72 phone bill, and you pay 5/6 of it

   1. One sixth: 72 ÷ 6 = 12.
   2. Five sixths: 12 × 5 = 60.
   3. $60 is less than $72, as it should be.

   Answer $60

   Next step: Dividing ↓
4. 4

   Dividing: keep, change, flip

   To divide by a fraction, multiply by its reciprocal: the same fraction turned upside down. The reciprocal of 2/3 is 3/2. The reciprocal of 5 (that is, 5/1) is 1/5.

   Keepthe first fraction as it is5/6

   Changethe ÷ sign to ××

   Flipthe second fraction3/2

   Only the second fraction flips: the one you divide by. Then multiply straight across and cancel, as before.

   1. 5/6 ÷ 2/3
   2. 5/6 × 3/2
   3. 5/2 × 1/2
   4. 5/4 = 1 1/4

   Shortcut

   Same bottom? Divide the tops

   When the two fractions have the same denominator, the answer is top ÷ top. 6/7 ÷ 2/7 asks how many 2-sevenths fit in 6-sevenths: 3.

   Next step: Why it works ↓
5. 5

   Why flipping works: dividing asks “how many fit?”

   12 ÷ 3 asks: how many 3s fit in 12? Dividing by a fraction asks the same question. 3 ÷ 1/4 asks: how many quarters fit in 3 wholes?

   Every whole holds 4 quarters, so you multiply by 4, which is 1/4 flipped. Dividing by 2/3 works the same way: every whole holds one and a half pieces of size 2/3, so you multiply by 3/2.

   “of”×1/2 of 30 → 1/2 × 30

   “how many … fit in”÷scoops of 1/2 cup in 4 cups → 4 ÷ 1/2

   “split into … equal parts”÷6 pounds split 4 ways → 6 ÷ 4

   Example

   How many 1/2-cup scoops are in 4 cups?

   1. Every cup holds 2 half-cups.
   2. 4 cups hold 4 × 2 = 8 half-cups.
   3. With the rule: 4 ÷ 1/2 = 4/1 × 2/1 = 8.

   Answer 8 scoops

   Next step: Mixed numbers ↓
6. 6

   Mixed numbers: turn them into fractions first

   A mixed number is a whole number and a fraction together, like 2 1/2 (two and one half). You cannot multiply or divide it piece by piece. First turn it into an improper fraction: a fraction whose top is as big as its bottom or bigger.

   2 1/2 → (2 × 2 + 1)/2 = 5/2

   Multiply the whole number by the bottom, add the top, and keep the same bottom. At the end, turn the answer back: see how many times the bottom goes into the top. That is the whole number, and what is left over goes on top.

   Example

   1 3/4 × 2 2/3

   First try1 × 2 = 2 and 3/4 × 2/3 = 1/2, so 2 1/2

   1. Convert: 1 3/4 = 7/4 and 2 2/3 = 8/3.
   2. 7/4 × 8/3: the 4 and the 8 cancel (both divide by 4), leaving 7/1 × 2/3.
   3. 7 × 2 = 14 on top, 3 on the bottom: 14/3.
   4. Back to a mixed number: 3 goes into 14 four times (12), with 2 left over: 4 2/3.

   Answer 4 2/3

   Example

   4 1/2 ÷ 1 1/2

   1. Convert: 9/2 ÷ 3/2.
   2. Keep, change, flip: 9/2 × 2/3.
   3. The 2s cancel, and the 9 and the 3 cancel (both divide by 3): 3/1 × 1/1 = 3.
   4. Check: three pieces of 1 1/2 make 4 1/2.

   Answer 3

   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, so you can spot it in the choices.

   Before you calculate, decide: is the question taking a part (“of”, multiply) or asking how many fit (divide)? Then check the size of your answer against the numbers in the question.

   Example · Easy

   You run for 2/3 of a 45-minute workout. How many minutes do you run?

   First try45 ÷ 2 × 3 = 67 1/2 minutes

   1. 67 1/2 is more than 45. Taking 2/3 of something must give less than the whole, so this try divided by the top and multiplied by the bottom: the wrong way round.
   2. Divide by the bottom: 45 ÷ 3 = 15. That is one third.
   3. Multiply by the top: 15 × 2 = 30. That is two thirds.

   Answer 30 minutes

   Example · Medium

   A cook has 9 cups of rice. Each batch of soup uses 3/4 cup. How many batches can the cook make?

   First try9 × 3/4 = 27/4 = 6 3/4 batches

   1. Each batch uses less than one cup, so there must be more than 9 batches. 6 3/4 is fewer: the try multiplied when it should divide.
   2. “How many 3/4 cups fit in 9 cups” is 9 ÷ 3/4.
   3. Keep, change, flip: 9/1 × 4/3. The 9 and the 3 cancel (both divide by 3): 3/1 × 4/1.
   4. 3 × 4 = 12 batches.

   Answer 12 batches

   Example · Test level

   A 40-foot rope is used at a campsite. 3/8 of it is cut off for a tent line. The rest is cut into pieces 2 1/2 feet long. How many pieces are there?

   First try3/8 of 40 = 15, and 15 ÷ 2 1/2 = 6 pieces

   1. 15 feet is the piece that was cut off. The question is about the rest of the rope.
   2. Cut off: 3/8 of 40. 40 ÷ 8 = 5, and 5 × 3 = 15 feet.
   3. The rest: 40 − 15 = 25 feet.
   4. Pieces: 25 ÷ 2 1/2 = 25/1 ÷ 5/2 = 25/1 × 2/5.
   5. The 25 and the 5 cancel (both divide by 5): 5 × 2 = 10 pieces.

   Answer 10 pieces

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.