Parts of parts

The idea

a/b × c/d = ac/bd (b ≠ 0, d ≠ 0), and a/b ÷ c/d = a/b × d/c = ad/bc (b ≠ 0, d ≠ 0, c ≠ 0).

A share of a share

A farmer owns 3/4 of a field. She gives 2/3 of that part to her daughter. How much of the whole field is that?

Will the daughter's share be bigger or smaller than 3/4 of the field?

• Bigger than 3/4
• Smaller than 3/4
• Exactly 3/4

Grid and chunks

Multiplying. The grid is one whole. Tap the column numbers to shade a/b of the width and the row numbers to shade c/d of the height. The cells shaded twice are the product.

Dividing. How many chunks of c/d fit into a/b? Add chunks until they fill the shaded part exactly.

ac/bd and ad/bc

a/b × c/d = ac/bd (b ≠ 0, d ≠ 0), and a/b ÷ c/d = a/b × d/c = ad/bc (b ≠ 0, d ≠ 0, c ≠ 0).

To multiply, multiply the tops and multiply the bottoms. To divide, turn the second fraction upside down and multiply.

Notes

a/b × c/d = ac/bd (b ≠ 0, d ≠ 0). a/b ÷ c/d = a/b × d/c = ad/bc (b ≠ 0, d ≠ 0, c ≠ 0).

Check yourself

What is 3/4 × 2/5 in lowest terms?

Which is the correct way to work out 3/4 ÷ 1/2?

How many 1/4-litre glasses can you fill from 3 litres? 3 ÷ 1/4 = 3/1 × 4/1 = ?

Answer: 12

3 × 4 = 12 glasses.

In a/b ÷ c/d, which value is not allowed?

• 3/10 — correct. 3 × 2 = 6 and 4 × 5 = 20, so the product is 6/20, which is 3/10 in lowest terms.
• 5/9. That adds the tops and the bottoms. To multiply, multiply them.
• 6/9. The bottoms are multiplied too: 4 × 5 = 20, not 9.
• 3/4 × 1/2 = 3/8. You must turn 1/2 upside down before multiplying.
• 4/3 × 1/2 = 4/6. Only the fraction you divide by is turned upside down, not the first one.
• 3/4 × 2/1 = 6/4 — correct. Yes. Turn the second fraction upside down and multiply. 6/4 = 3/2.
• c = 0 — correct. Yes. c/d would be 0, and we cannot divide by 0.
• a = 0. a = 0 is fine: 0 ÷ c/d = 0.
• d = 1. d = 1 is fine: c/1 is just c.