Simplest form

The idea

A ratio is reduced to its simplest form by dividing both its terms by their HCF.

60 : 40 with smaller numbers

Picture A is 60 mm wide and 40 mm tall. The ratio of width to height is 60 : 40. Can we say the same thing with smaller numbers?

Cut 60 mm and 40 mm into equal groups with nothing left over. What is the biggest group size that works?

• 10 mm
• 20 mm
• 40 mm

Cut both bars into groups

Pick a group size. Both bars are cut into groups of that size. A red piece means something is left over, so that size does not fit.

Divide by the HCF

A ratio is reduced to its simplest form by dividing both its terms by their HCF.

The HCF of 60 and 40 is 20. 60 ÷ 20 = 3 and 40 ÷ 20 = 2, so 60 : 40 becomes 3 : 2.

Dividing by a smaller common factor, like 10, gives 6 : 4. That still has a common factor, 2, so it is not the simplest form.

Notes

A ratio is reduced to its simplest form by dividing both its terms by their HCF.

Check yourself

What is 18 : 24 in its simplest form?

What is the HCF of 72 and 96?

Answer: 24

72 = 2³ × 3² and 96 = 2⁵ × 3. They share 2 × 2 × 2 × 3 = 24. So 72 : 96 = 3 : 4.

Picture E is 60 mm wide and 60 mm tall. What is the ratio 60 : 60 in its simplest form?

What is the HCF of 3 and 4? Then you know whether 3 : 4 can be reduced.

Answer: 1

3 and 4 share only the factor 1. Dividing by 1 changes nothing, so 3 : 4 is already in its simplest form.

• 6 : 8. 18 and 24 were divided by 3, but 6 and 8 are both even. It can be reduced further.
• 3 : 4 — correct. Yes! The HCF of 18 and 24 is 6, and 18 ÷ 6 = 3, 24 ÷ 6 = 4.
• 9 : 12. 18 and 24 were divided by 2, but 9 and 12 are both multiples of 3. It can be reduced further.
• 60 : 60. The terms have the common factor 60, so the ratio can be reduced.
• 2 : 2. Both terms can be divided by more than 30. The HCF of 60 and 60 is 60.
• 1 : 1 — correct. Yes! The HCF of 60 and 60 is 60, so 60 ÷ 60 = 1 and 60 ÷ 60 = 1.