Mathematics • Year 8 • Unit 1 • Lesson 17
Sharing It Out — Ratio in Real Life
Split lottery wins, restaurant bills, inheritance, recipes and group gifts using the unitary method. Then explain why "share equally" and "share in a ratio" are different things.
1. Word problems
Each problem uses the lesson's idea: sum of parts, value of 1 part, multiply for each share, sum check. Show your working — a final answer with no working only earns half marks.
1.1 — Lottery win. Three friends pooled money to buy a lottery ticket and won $480. They agreed beforehand to split any winnings in the ratio 1 : 2 : 3 because of how much each had contributed.
(a) How much does each friend receive?
(b) Show your sum check. 3 marks
1.2 — Group gift. Mia, Jay and Tahlia bought a $96 birthday gift for their friend. They agreed to contribute in the ratio 3 : 4 : 5 because their part-time pay is different.
(a) How much does each pay?
(b) Check the three contributions add to $96. 3 marks
1.3 — Concrete bag. A 45 kg bag of pre-mixed dry concrete is in the ratio 2 : 3 : 4 (cement : sand : gravel).
(a) How much of each ingredient is in the bag?
(b) If you only need 15 kg of total mix for a small job, how much cement should you use? 3 marks
1.4 — Reverse problem. Lucia and Asha split $120 in the ratio 7 : 5. Lucia gets the bigger share.
(a) How much does Lucia receive?
(b) Asha now realises she actually contributed more than her share suggests. If the ratio were instead 5 : 7 (Asha getting the bigger share), how much would she get? 3 marks
1.5 — Inheritance. Three siblings inherit $420 and split it in the ratio 2 : 3 : 5 (in proportion to how much they had contributed to a family business).
(a) Calculate each share.
(b) The middle sibling argues "we should just split it equally, $140 each". By how much would the OLDEST sibling (who got the 5-part share) lose if they switched to an equal split? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says "$480 split in the ratio 1 : 2 : 3 just means three equal shares of $160 each — there are three people, so divide by three." In your own words, explain (i) why this is wrong, (ii) what the correct shares are, and (iii) what "1 : 2 : 3" actually means in plain English. Use the phrase "the shares are unequal" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Lottery win ($480 in 1 : 2 : 3)
(a) Total parts = 1 + 2 + 3 = 6. 1 part = $480 ÷ 6 = $80. Shares: $80, $160, $240.
(b) Check: $80 + $160 + $240 = $480 ✓.
1.2 — Group gift ($96 in 3 : 4 : 5)
(a) Total parts = 12. 1 part = $96 ÷ 12 = $8. Mia: $24; Jay: $32; Tahlia: $40.
(b) Check: $24 + $32 + $40 = $96 ✓.
1.3 — Concrete bag
(a) Total parts = 9. 1 part = 45 ÷ 9 = 5 kg. Cement: 10 kg; sand: 15 kg; gravel: 20 kg. Check: 10 + 15 + 20 = 45 ✓.
(b) For 15 kg total: 1 part = 15 ÷ 9 = 5/3 kg ≈ 1.67 kg. Cement = 2 parts = 10/3 kg ≈ 3.33 kg (about 3 1/3 kg).
1.4 — Reverse problem ($120 in 7 : 5)
(a) Total parts = 12. 1 part = $120 ÷ 12 = $10. Lucia (7 parts) = $70.
(b) If ratio is 5 : 7 with Asha now getting the bigger 7-part share, Asha = 7 × $10 = $70. (The bigger share is always 7 × $10 = $70, no matter who is named.)
1.5 — Inheritance ($420 in 2 : 3 : 5)
(a) Total parts = 10. 1 part = $420 ÷ 10 = $42. Shares: $84, $126, $210. Check: 84 + 126 + 210 = $420 ✓.
(b) Equal split = $420 ÷ 3 = $140 each. Oldest currently gets $210, so would lose $70 ($210 − $140).
2.1 — Explain your thinking (sample response)
The classmate is wrong because "1 : 2 : 3" doesn't mean three equal shares — it means the shares are unequal. The three numbers tell us the RELATIVE sizes: the second person gets twice as much as the first, and the third person gets three times as much. To split correctly, add the ratio numbers (1 + 2 + 3 = 6 parts), find what 1 part is worth ($480 ÷ 6 = $80), then multiply: $80, $160, $240. Dividing $480 by 3 would only be right if the ratio were 1 : 1 : 1 (three equal shares).
Marking: 1 mark for spotting the mistake (treating ratio as equal); 1 mark for the correct shares ($80, $160, $240); 1 mark for explaining what "1 : 2 : 3" means in plain English; 1 mark for a clear, full-sentence explanation that uses "the shares are unequal".