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Multiplying fractions

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Fractions

Multiplying fractions

Straight across: tops together, bottoms together. It is the easiest of the four operations — and the one people expect to be harder.

A fraction OF a fraction is the overlap3/5 across1/23/5 × 1/2= 3/10tops together, bottoms togethermultiplying by a proper fraction always makes it SMALLER
The word “of” means multiply.

The rule

To multiply fractions, multiply the NUMERATORS together and multiply the DENOMINATORS together. 2/5 × 7/8 = (2 × 7)/(5 × 8) = 14/40 Then cancel to the simplest form. Both 14 and 40 divide by 2: 14/40 = 7/20

Why it works

Take a rectangle and shade 3/5 of it going across. Now shade 1/2 of THAT going down. The part shaded both ways is 3/10 of the whole rectangle — and 3/5 × 1/2 = 3/10. Multiplying fractions is finding a fraction OF a fraction. The word 'of' means multiply.

Fraction of a fraction

In a swimming club, 2/3 of the members are children, and 3/5 of the children are boys. What fraction of the club are boys? '3/5 of 2/3' means 3/5 × 2/3 = 6/15 = 2/5 So 2/5 of the club members are boys. Notice the answer is smaller than either fraction — multiplying by a proper fraction always makes things smaller.

Worked examples

2/5 × 7/8

  1. Multiply the numerators: 2 × 7 = 14.
  2. Multiply the denominators: 5 × 8 = 40.
  3. Cancel: both divide by 2.

Answer: 7/20

3/5 × 1/2 with a picture

  1. Shade 3/5 across a rectangle.
  2. Shade 1/2 of that going down.
  3. The doubly shaded part is 3/10.

Answer: 3/10

3/5 of 2/3

  1. 'Of' means multiply.
  2. 3/5 × 2/3 = 6/15.
  3. Cancel by 3.

Answer: 2/5

Why the answer got smaller

  1. You are taking a fraction OF a fraction.
  2. Multiplying by a proper fraction always shrinks things.

Answer: It should be smaller than either fraction

1/2 × 1/2

  1. 1 × 1 = 1 and 2 × 2 = 4.

Answer: 1/4 — half of a half

Squaring 2/3

  1. To square means multiply by itself.
  2. 2 × 2 = 4 and 3 × 3 = 9.

Answer: 4/9

Where you see this in real life

  • Half of a quarter of a cake is 1/2 × 1/4 = 1/8 of the whole cake.
  • A 3/4-full tank that you use 2/3 of: you have used 2/3 × 3/4 = 1/2 of a full tank.

Key words

To square
to multiply a number by itself
Cancel
divide top and bottom by a common factor
Simplest form
a fraction that will not cancel any further
Numerator
the top number of a fraction
Denominator
the bottom number of a fraction
Proper fraction
a fraction smaller than 1
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