Multiplying fractions

Half of three quarters is three eighths: 1/2 × 3/4 = 3/8, and the area model shows all 8 pieces.

A whole number times a fraction

Start with what multiplication already means: repeated addition. 3 × 2/5 is three lots of 2/5, which is 2/5 + 2/5 + 2/5 = 6/5. Only the numerator changed, because you are counting more fifths. So a whole number times a fraction multiplies the numerator: 3 × 2/5 = (3 × 2)/5 = 6/5.

Multiplying a whole number by a fraction is also the same as taking a fraction of it, which you met in unit 1. 5/6 × 12 asks for 5/6 of 12: 12 ÷ 6 = 2, and 2 × 5 = 10. In word problems, the word of almost always means multiply.

A fraction of a fraction

What is half of 3/4? Draw a square and split it into 4 columns. Shade 3 of them for 3/4. Now split the square into 2 rows and take the top row, half of everything. The part that is shaded both ways covers 3 small cells, and the whole square now has 4 × 2 = 8 cells. So half of 3/4 is 3/8.

The picture explains the rule. The denominators multiply because cutting into 4 columns and then 2 rows makes 4 × 2 small cells. The numerators multiply because the overlap is 3 columns wide and 1 row tall, 3 × 1 cells. So a/b × c/d = (a × c)/(b × d): multiply the numerators, multiply the denominators. No common denominator is needed.

Simplify before you multiply

Products can get large quickly. 4/9 × 3/8 = 12/72, which then simplifies to 1/6. It is easier to cancel common factors first. Any numerator may be cancelled with any denominator, even across the two fractions, because all four numbers end up in one big fraction (4 × 3)/(9 × 8). The 4 and the 8 share 4, leaving 1 and 2. The 3 and the 9 share 3, leaving 1 and 3. What remains is 1/3 × 1/2 = 1/6.

Why the answer can be smaller

With whole numbers, multiplying makes things bigger. With fractions less than 1 it makes them smaller, which surprises many people. 7/8 × 5 is less than 5, because you are taking 7/8 of 5, not all of it. Multiplying by a number less than 1 shrinks, multiplying by 1 changes nothing, and multiplying by a number greater than 1 enlarges.

A fraction of a fraction in everyday problems

Multiplying fractions answers questions about part of a part. Suppose 2/3 of the students in a school take music, and 1/4 of those play the drums. The drummers are 1/4 of 2/3 of the school: 1/4 × 2/3 = 2/12 = 1/6 of all the students. The order does not matter, since 2/3 × 1/4 gives the same 1/6.

Squaring a fraction is multiplying it by itself. A square tile with sides of 2/3 m has area 2/3 × 2/3 = 4/9 square metres. The area, 4/9, is smaller than the side length, 2/3, which would be impossible with whole numbers bigger than 1.

Mixed numbers: convert first

Mixed numbers must be turned into improper fractions before multiplying. 1 1/2 × 2 2/3 = 3/2 × 8/3. Cancel the 3s and the 2 with the 8: 1/1 × 4/1 = 4. A common mistake is to multiply the wholes and the fractions separately, 1 × 2 plus 1/2 × 2/3, which gives 2 1/3. That leaves out the cross terms, 1 × 2/3 and 1/2 × 2, the same way that 12 × 13 is not 1 × 1 plus 2 × 3.

Worked examples

Example 1. A recipe needs 2/3 cup of sugar and you are making 3/4 of the recipe. 3/4 × 2/3 = 6/12 = 1/2 cup.

Example 2. A rectangle of card is 3/4 m long and 2/5 m wide. Its area is 3/4 × 2/5 = 6/20 = 3/10 square metres.

Example 3. 2 1/4 × 4/3 = 9/4 × 4/3. Cancel the 4s and reduce 9 with 3: 3/1 × 1/1 = 3.