Simplest form of a multi-term ratio

The idea

A ratio with several terms is reduced to its simplest form, as for two terms, by dividing all the terms by their HCF.

Can this ratio be made simpler?

40 students got grades A to E: 12, 10, 8, 6 and 4 students. The ratio of these five numbers is 12 : 10 : 8 : 6 : 4.

What is the simplest form of this ratio?

• 3 : 2.5 : 2 : 1.5 : 1
• 6 : 5 : 4 : 3 : 2
• It cannot be made simpler

Cut every bar into groups

Each term is a bar. Choose a group size and cut all the bars into groups. A bar with a red piece left over does not fit. The number of groups in each bar is the new term.

Divide all the terms by their HCF

A ratio with several terms is reduced to its simplest form, as for two terms, by dividing all the terms by their HCF.

The HCF of 12, 10, 8, 6, 4 is 2. Divide every term by 2: 12 : 10 : 8 : 6 : 4 = 6 : 5 : 4 : 3 : 2

The ratio is in simplest form when the HCF of all its terms is 1. For 20 : 30 : 15 the HCF is 5, so it becomes 4 : 6 : 3. The terms 4 and 6 share the factor 2, but 3 does not, so no bigger group fits all three bars.

Notes

A ratio with several terms is reduced to its simplest form, as for two terms, by dividing all the terms by their HCF.

Check yourself

Reduce 18 : 24 : 30 to its simplest form. What is the last term?

Answer: 5

The HCF of 18, 24 and 30 is 6. Dividing every term by 6 gives 3 : 4 : 5, so the last term is 5.

Which is the simplest form of 20 : 30 : 15?

Is 4 : 6 : 3 in simplest form?

A ratio in simplest form is 3 : 4 : 5. It was made by dividing by an HCF of 7. What was the last term of the original ratio?

Answer: 35

Undo the division by multiplying by 7: 3 : 4 : 5 becomes 21 : 28 : 35. The last term is 35.

• 2 : 3 : 1.5. That divides by 10, but 10 does not divide 15, so a term is not a whole number. The HCF is 5.
• 4 : 6 : 3 — correct. Yes! The HCF of 20, 30 and 15 is 5, and 20 : 30 : 15 = 4 : 6 : 3.
• 10 : 15 : 7.5. That divides by 2, but 2 does not divide 15. The HCF of all three terms is 5.
• 20 : 30 : 15. It can be reduced: 5 divides all three terms.
• No, because 4 and 6 are both even.. Dividing by 2 would turn 3 into 1.5, which is not a whole number. All three terms must share the factor.
• No, because 6 is a multiple of 3.. But 4 is not a multiple of 3. A bigger group must fit all three terms.
• Yes, because the HCF of 4, 6 and 3 is 1. — correct. Yes! No number bigger than 1 divides all three terms, so it cannot be reduced.