Mathematics • Year 8 • Unit 1 • Lesson 16

Simplifying Ratios

Reduce ratios to lowest terms by dividing every part by the HCF. One worked example, one guided example with blanks, then eight independent problems including three-part ratios and mixed units.

Build · I Do / We Do / You Do

1. I do — fully worked example

Watch every step. Each line has a short reason so you can see why we do it, not just what we do.

Problem. Simplify 24 : 36 to its lowest terms.

Step 1 — List the factors of each number.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Reason: we need the biggest number that divides BOTH parts of the ratio.

Step 2 — Pick the Highest Common Factor (HCF).

HCF(24, 36) = 12

Reason: 12 is the biggest number on BOTH lists of factors.

Step 3 — Divide BOTH parts by the HCF.

24 ÷ 12 : 36 ÷ 12 = 2 : 3

Reason: dividing both parts by the same number keeps the ratio equivalent.

Step 4 — Check the result is fully simplified.

2 and 3 share no common factor greater than 1. Done.

Reason: if you can still divide, you haven't finished simplifying.

Answer: 24 : 36 = 2 : 3.

Stuck? Revisit lesson § Card 5 — "Simplifying Two-Part Ratios".

2. We do — fill in the missing steps

Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks

Problem. Simplify 30 : 45 to its lowest terms.

Step 1 — Find the HCF. HCF(30, 45) = ______

Step 2 — Divide both parts by the HCF:

30 ÷ ______ : 45 ÷ ______ = ______ : ______

Step 3 — Check no common factor > 1 remains:

Common factor of ______ and ______ is ______. (Should be 1.)

Step 4 — Final answer:

30 : 45 = ______ : ______

Stuck? Both 30 and 45 are in the 15 times table.

3. You do — independent practice

Show your working in the space under each problem. The first four are foundation (two-part ratios with small HCFs). The middle two are standard (three-part ratios or larger HCFs). The last two are extension (mixed units — convert first).

Foundation — two-part simplifying

3.1 Simplify 20 : 30. 1 mark

3.2 Simplify 16 : 24. 1 mark

3.3 Simplify 45 : 60. 1 mark

3.4 Simplify 36 : 54. 1 mark

Standard — three-part ratios

3.5 Simplify 9 : 27 : 36. (Hint: find the HCF of all THREE numbers.) 2 marks

3.6 Simplify 10 : 25 : 40. 2 marks

Extension — mixed units (convert first!)

3.7 Simplify $3 : 50 cents. (Hint: turn dollars into cents first — $3 = 300 cents.) 2 marks

3.8 Simplify 40 minutes : 2 hours. (Hint: convert hours into minutes first.) 2 marks

Stuck on 3.7 / 3.8? Revisit lesson § Card 6 — "Three-Part Ratios and Mixed Units". The golden rule: SAME UNITS first, simplify second.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (faded 30 : 45)

Step 1: HCF(30, 45) = 15.
Step 2: 30 ÷ 15 : 45 ÷ 15 = 2 : 3.
Step 3: common factor of 2 and 3 is 1 — fully simplified.
Step 4: 30 : 45 = 2 : 3.

3.1 — 20 : 30

HCF(20, 30) = 10. 20 ÷ 10 : 30 ÷ 10 = 2 : 3.

3.2 — 16 : 24

HCF(16, 24) = 8. 16 ÷ 8 : 24 ÷ 8 = 2 : 3.

3.3 — 45 : 60

HCF(45, 60) = 15. 45 ÷ 15 : 60 ÷ 15 = 3 : 4.

3.4 — 36 : 54

HCF(36, 54) = 18. 36 ÷ 18 : 54 ÷ 18 = 2 : 3.

3.5 — 9 : 27 : 36

HCF(9, 27, 36) = 9. 9 ÷ 9 : 27 ÷ 9 : 36 ÷ 9 = 1 : 3 : 4.

3.6 — 10 : 25 : 40

HCF(10, 25, 40) = 5. 10 ÷ 5 : 25 ÷ 5 : 40 ÷ 5 = 2 : 5 : 8.

3.7 — $3 : 50 cents

Convert: $3 = 300 cents. Now both sides are in cents: 300 : 50. HCF = 50. 300 ÷ 50 : 50 ÷ 50 = 6 : 1.

3.8 — 40 min : 2 hours

Convert: 2 hours = 120 minutes. Now both in minutes: 40 : 120. HCF = 40. 40 ÷ 40 : 120 ÷ 40 = 1 : 3.