Mathematics • Year 8 • Unit 1 • Lesson 15
Ratios — Mixed Challenge
Pull together everything from Lesson 15: writing a ratio, simplifying with HCF, mixed-unit ratios, three-part ratios, total parts and fractions of a whole. 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 15. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 Simplify these ratios: (a) 18 : 24, (b) 35 : 14, (c) 50 : 100.
1.2 A bag has marbles in the ratio red : blue : green = 2 : 3 : 5. There are 30 marbles in the bag. Find the number of each colour.
1.3 Write the ratio 750 g : 1.5 kg in simplest form. (Hint: convert first.)
1.4 In a 3 : 1 flour-to-sugar recipe, how much flour and how much sugar are needed if you want to make a total of 600 g of mix?
1.5 A school camp has 90 students in the ratio Year 7 : Year 8 : Year 9 = 4 : 3 : 2. How many students from each year?
1.6 Classify each item as RATIO, RATE or FRACTION: (a) $\tfrac{2}{5}$ of the class wears glasses; (b) 80 km in 2 hours; (c) a recipe of 2 : 1 flour to butter; (d) $\$3.40$/L of milk; (e) 12 girls : 18 boys.
2. Find the mistake
Another student is trying to share $\$60$ between two friends in the ratio 2 : 3. Their working is shown below. Exactly one line contains the key mistake. Spot it, explain why it's wrong, then re-do the calculation correctly. 3 marks
Student's working — share $\$60$ in the ratio 2 : 3:
Line 1: Ratio is 2 : 3.
Line 2: First friend gets 2/3 of $\$60$ and the second friend gets 3/3 (= all) of $\$60$.
Line 3: First friend: 2/3 × $\$60$ = $\$40$.
Line 4: Second friend: 3/3 × $\$60$ = $\$60$.
(a) Which line contains the key mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: check the totals.)
(c) Re-do the share correctly. State the total parts, the dollars per part, and what each friend actually receives.
Stuck? In a ratio a : b, the denominators of the fractions are (a + b), not b. So 2 : 3 gives fractions 2/5 and 3/5 — not 2/3 and 3/3.3. Open-ended challenge — design a three-part recipe
This question has more than one valid answer. 4 marks
3.1 Design a three-part recipe / mix where:
- the three parts have three DIFFERENT whole-number values (e.g., 2 : 3 : 5, but not 2 : 2 : 4),
- the total parts add to a number that divides 600 cleanly (so it scales to 600 g),
- and you can name a realistic context (smoothie, fruit salad, dog food blend, paint mix, etc.).
Write:
(i) Your context and the three-part ratio.
(ii) The total parts in your ratio.
(iii) The amount (in g) of each ingredient for a 600 g batch.
(iv) The fraction of the mix that each ingredient represents.
Bonus: Scale your recipe to make 1.5 kg of the mix — show the new amounts of each ingredient.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Simplify ratios
(a) 18 : 24, HCF = 6, → 3 : 4.
(b) 35 : 14, HCF = 7, → 5 : 2.
(c) 50 : 100, HCF = 50, → 1 : 2.
1.2 — Marble bag 2 : 3 : 5, total 30
Total parts = 2 + 3 + 5 = 10. Per part = 30 ÷ 10 = 3. Red = 2 × 3 = 6; blue = 3 × 3 = 9; green = 5 × 3 = 15. (Check: 6 + 9 + 15 = 30 ✓.)
1.3 — 750 g : 1.5 kg
Convert 1.5 kg = 1500 g. Ratio 750 : 1500. HCF = 750: 1 : 2.
1.4 — Flour : sugar 3 : 1 in 600 g
Total parts = 4. Per part = 150 g. Flour = 3 × 150 = 450 g; sugar = 1 × 150 = 150 g.
1.5 — Camp 4 : 3 : 2 in 90 students
Total parts = 9; per part = 10 students. Year 7 = 40; Year 8 = 30; Year 9 = 20. (Check: 40 + 30 + 20 = 90 ✓.)
1.6 — Classify each
(a) FRACTION (part of a whole).
(b) RATE (km / h, different units — gives 40 km/h).
(c) RATIO (flour : butter, same kind of thing).
(d) RATE ($ / L, different units).
(e) RATIO (12 : 18 simplifies to 2 : 3 girls : boys).
2 — Find the mistake
(a) The key mistake is on Line 2 (and it then breaks Lines 3 and 4).
(b) The student used the wrong denominators. In a ratio a : b, the fractions of the whole are a/(a+b) and b/(a+b), not a/b and b/b. The student's working gives $\$40 + \$60 = \$100$, which is more than the $\$60$ being shared — the dollars don't add up, which is the clue that something is wrong.
(c) Corrected: total parts = 2 + 3 = 5. Per part = $\$60 \div 5 = \$12$. First friend = 2 × $\$12 = \textbf{\$24}$; second friend = 3 × $\$12 = \textbf{\$36}$. (Check: $24 + 36 = \$60$ ✓.) Equivalent fractions: 2/5 and 3/5 of the $\$60$.
3 — Open-ended challenge (sample solution)
(i) Context: A smoothie made of banana : yoghurt : milk in the ratio 1 : 2 : 3.
(ii) Total parts = 1 + 2 + 3 = 6.
(iii) For 600 g: per part = 600 ÷ 6 = 100 g. Banana = 100 g; yoghurt = 200 g; milk = 300 g. (Check: 100 + 200 + 300 = 600 g ✓.)
(iv) Fractions of the mix: banana = 1/6; yoghurt = 2/6 = 1/3; milk = 3/6 = 1/2.
Bonus: Scaled to 1.5 kg (1500 g): per part = 1500 ÷ 6 = 250 g. Banana = 250 g; yoghurt = 500 g; milk = 750 g. (Check: 250 + 500 + 750 = 1500 g ✓.)
Marking: 1 mark for a valid three-different-part ratio with a clean total; 1 mark for correct ingredient amounts in 600 g; 1 mark for the fraction breakdown; 1 bonus mark for a correctly scaled 1.5 kg version.