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.