Mathematics • Year 7 • Unit 1 • Lesson 12
Ratios — Mixed Challenge
Pull everything from Lesson 12 together: simplify ratios, divide amounts in a ratio, find missing values, work with three-part ratios, and spot a classic "different units" mistake. Finish with an open-ended puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 12. Decide which method applies before you start. Show working. 2 marks each
1.1 Simplify the ratio 24:36.
1.2 Share $150 in the ratio 3:7. List both shares.
1.3 Find x if 4:7 = x:35.
1.4 Three friends invest in a business in the ratio 2:3:5. The total investment is $24,000. How much did each friend invest?
1.5 Simplify the ratio 0.5 : 1.5 (multiply both sides to clear the decimals first).
1.6 A bag has red and blue marbles in the ratio 3:5. The total number of marbles is 24. How many red marbles are there, and what fraction of the bag is red?
2. Find the mistake
Another Year 7 student tried to simplify the ratio 30 cm : 1.2 m. Their working is below. Exactly one line contains the mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — simplify 30 cm : 1.2 m:
Line 1: Ratio is 30 : 1.2.
Line 2: Multiply both by 10 to clear the decimal: 300 : 12.
Line 3: HCF(300, 12) = 12. Divide both by 12.
Line 4: 300 ÷ 12 = 25 and 12 ÷ 12 = 1.
Line 5: Simplified ratio: 25 : 1.
(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? The two numbers have different units. You must convert to the same unit before writing the ratio — 1.2 m is not just "1.2" when next to 30 cm.3. Open-ended challenge — design a ratio
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Your job is to design a fruit punch for a party using orange juice, pineapple juice and soda water in some ratio. The rules are:
(i) The punch must be a 3-part ratio using only whole numbers.
(ii) The total mixture must be exactly 12 L.
(iii) There must be more orange juice than pineapple juice, and more pineapple juice than soda water (so the ratio numbers must decrease: orange > pineapple > soda).
(iv) Each ingredient must be at least 1 L.
Write down your ratio, then calculate how many litres of each ingredient. Show your working and check the totals.
Bonus: How many different valid ratios can you find that satisfy all four rules? List them all and explain how you know you have them all.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Simplify 24:36
HCF(24, 36) = 12. 24 ÷ 12 = 2, 36 ÷ 12 = 3. Answer: 2:3.
1.2 — $150 in ratio 3:7
Total parts = 10. One part = $150 ÷ 10 = $15. Shares: 3 × $15 = $45 and 7 × $15 = $105. Check: $45 + $105 = $150 ✓.
1.3 — 4:7 = x:35
Cross-multiply: 4/7 = x/35 → 7x = 4 × 35 = 140 → x = 20. Check: 4:7 = 20:35 (× 5 both sides) ✓.
1.4 — Business investment 2:3:5 for $24,000
Total parts = 10. One part = $24,000 ÷ 10 = $2,400.
Friend 1 = 2 × $2,400 = $4,800. Friend 2 = 3 × $2,400 = $7,200. Friend 3 = 5 × $2,400 = $12,000.
Check: $4,800 + $7,200 + $12,000 = $24,000 ✓.
1.5 — Simplify 0.5 : 1.5
Multiply both sides by 2 to clear the decimals: 1 : 3. (Or × 10: 5 : 15, then ÷ 5 to get 1 : 3.) Answer: 1:3.
1.6 — Red and blue marbles 3:5, total 24
Total parts = 3 + 5 = 8. One part = 24 ÷ 8 = 3 marbles. Red = 3 × 3 = 9 marbles. Red as a fraction of total = 3/8 of the bag. (Check: 3/8 of 24 = 9 ✓.)
2 — Find the mistake
(a) The mistake is on Line 1.
(b) The student wrote "30 : 1.2" without converting the units. 30 cm and 1.2 m are in different units — you must convert to a common unit first.
(c) Corrected working:
Line 1 (fixed): Convert 1.2 m to cm: 1.2 m = 120 cm. So the ratio is 30 cm : 120 cm = 30 : 120.
Line 2 (new): HCF(30, 120) = 30. Divide both by 30: 30 ÷ 30 = 1, 120 ÷ 30 = 4.
Line 3 (new): Simplified ratio: 1 : 4.
Sanity check: 30 cm is much smaller than 1.2 m (= 120 cm), so the ratio should be 1 to something bigger, not 25 to 1.
3 — Fruit punch puzzle (sample solutions)
We need O : P : S with O > P > S ≥ 1, all whole numbers, and the total (in parts) must divide evenly into 12.
Sample answer. Ratio 3:2:1 has 6 parts. One part = 12 ÷ 6 = 2 L. Orange = 6 L, pineapple = 4 L, soda = 2 L. Check: 6 + 4 + 2 = 12 L ✓; 6 > 4 > 2 ✓; each ≥ 1 L ✓.
All valid ratios. Total parts must divide 12 → possible totals: 6 or 12 (3 and 4 are too small for a strict 3-part decreasing ratio with each part ≥ 1).
Total = 6 parts: O > P > S ≥ 1, sum = 6 → only 3:2:1 works (6 parts).
Total = 12 parts: O > P > S ≥ 1, sum = 12 → possibilities: (9,2,1), (8,3,1), (7,4,1), (6,5,1), (8,2,2 fails — not strictly decreasing), (7,3,2), (6,4,2), (5,4,3). That's 7 valid ratios at 12 parts.
Together with 3:2:1 at 6 parts, there are 8 valid ratios in total.
We know we have them all because we systematically enumerated all sums = 6 and = 12 with strictly decreasing whole-number parts ≥ 1, and 6 and 12 are the only divisors of 12 that allow three such parts.
Marking: 2 marks for any valid ratio with correct litre breakdown and check; 1 mark for explaining how the "one part" was found; 1 bonus mark for systematic listing of all valid ratios (accept 8 or any partial list with clear reasoning).