Mathematics • Year 8 • Unit 1 • Lesson 16
Simplifying Ratios — Mixed Challenge
Pull together everything from Lesson 16: HCF for two-part and three-part ratios, mixed-unit conversions, and ratio-to-fraction thinking. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 16. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 Simplify 42 : 56.
1.2 Simplify 6 : 9 : 15. (Find the HCF of all three.)
1.3 Simplify 500 mL : 2 L.
1.4 Simplify 12 kg : 400 g. (Convert kg to g first.)
1.5 Two ratios are claimed to be equivalent: 4 : 6 and 14 : 21. Are they? Simplify each and compare.
1.6 A bag has red and blue marbles in the ratio 14 : 35. (a) Simplify the ratio. (b) What fraction of the bag is red?
2. Find the mistake
Another student has tried to simplify 12 : 18 : 24. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — simplify 12 : 18 : 24:
Line 1: HCF needed: factors of 12 = 1, 2, 3, 4, 6, 12; factors of 18 = 1, 2, 3, 6, 9, 18; factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24.
Line 2: The HCF of all three numbers is 3.
Line 3: Divide each part by 3: 12 ÷ 3 : 18 ÷ 3 : 24 ÷ 3 = 4 : 6 : 8.
Line 4: Simplest form: 4 : 6 : 8.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 4 — "Partial simplification". Look at the lists in Line 1: what's the BIGGEST number that appears on ALL three lists?3. Open-ended challenge — design your own concrete mix
This question has more than one valid answer. 4 marks
3.1 A "standard" concrete mix uses the ratio 2 : 3 : 4 (cement : sand : gravel). Imagine you are mixing a batch using this ratio.
(i) Choose any total mass between 30 kg and 100 kg for your batch. Write it down.
(ii) Work out how much cement, sand and gravel you'd need to keep the 2 : 3 : 4 ratio. (Hint: total parts = 9. Find what 1 part weighs first, then multiply.)
(iii) Check that your three masses simplify back to 2 : 3 : 4.
Bonus: What fraction of your batch is sand?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 42 : 56
HCF(42, 56) = 14. 42 ÷ 14 : 56 ÷ 14 = 3 : 4.
1.2 — 6 : 9 : 15
HCF(6, 9, 15) = 3. 6 ÷ 3 : 9 ÷ 3 : 15 ÷ 3 = 2 : 3 : 5.
1.3 — 500 mL : 2 L
Convert: 2 L = 2000 mL. So 500 : 2000. HCF = 500. 1 : 4.
1.4 — 12 kg : 400 g
Convert: 12 kg = 12 000 g. So 12 000 : 400. HCF = 400. 30 : 1.
1.5 — Are 4 : 6 and 14 : 21 equivalent?
4 : 6 = (÷ 2) = 2 : 3. 14 : 21 = (÷ 7) = 2 : 3. Both simplify to 2 : 3, so YES — the two ratios ARE equivalent.
1.6 — Marble bag 14 : 35
(a) HCF(14, 35) = 7. 2 : 5.
(b) Total parts = 2 + 5 = 7. Red = 2/7 of the bag.
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong HCF is then used in Line 3).
(b) The student picked 3 as the HCF, but 6 appears on ALL THREE lists in Line 1. The HCF should be the BIGGEST common factor, not just any common factor. So HCF = 6, not 3.
(c) Corrected working:
HCF(12, 18, 24) = 6.
Divide each part by 6: 12 ÷ 6 : 18 ÷ 6 : 24 ÷ 6 = 2 : 3 : 4.
Sanity check: 2, 3 and 4 share no common factor greater than 1, so fully simplified.
3 — Open-ended challenge (sample solution)
Sample: choose total = 45 kg. Total parts = 2 + 3 + 4 = 9. One part = 45 ÷ 9 = 5 kg.
Cement: 2 × 5 = 10 kg. Sand: 3 × 5 = 15 kg. Gravel: 4 × 5 = 20 kg.
Check: 10 + 15 + 20 = 45 kg ✓. Simplify 10 : 15 : 20 → HCF = 5 → 2 : 3 : 4 ✓.
Bonus: Sand = 3/9 = 1/3 of the batch.
Other valid totals: 36 kg (1 part = 4 kg → 8, 12, 16), 90 kg (1 part = 10 kg → 20, 30, 40), 99 kg (1 part = 11 kg → 22, 33, 44).
Marking: 1 mark for picking a valid total in the range; 1 mark for the correct three masses; 1 mark for the simplify-back check; 1 mark for the correct sand fraction (1/3).