Mathematics • Year 8 • Unit 1 • Lesson 17
Ratio Division — Mixed Challenge
Pull together everything from Lesson 17: two-part and three-part splits, reverse problems, and sum checks. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different mix of ideas from Lesson 17. Decide which move applies before you start writing. Show your working and ALWAYS do a sum check. 3 marks each
1.1 Split $700 in the ratio 3 : 4.
1.2 Split 45 kg in the ratio 1 : 2 : 6.
1.3 Split $360 in the ratio 5 : 3 : 2.
1.4 In a ratio split of 3 : 4 : 5 totalling 48 kg, find the middle share.
1.5 Reverse problem: Two people share a sum in the ratio 4 : 5. The bigger share is $200. Find the total.
1.6 Three friends agreed to split a $540 prize in the ratio 1 : 2 : 3. Find the smallest share, the largest share, and the difference between them.
2. Find the mistake
Another student has tried to split $540 in the ratio 1 : 2 : 3. 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 — split $540 in 1 : 2 : 3:
Line 1: Total parts = 1 + 2 + 3 = 6.
Line 2: Value of 1 part = $540 ÷ 3 = $180.
Line 3: Shares: 1 × $180 = $180, 2 × $180 = $360, 3 × $180 = $540.
Line 4: Final shares: $180, $360, $540.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: do the sum check — do the three shares add to $540?)
(c) Write out the corrected working in full, including the corrected final shares and a sum check.
Stuck? Look at the divisor in Line 2. Should we divide by the NUMBER OF PEOPLE or by the TOTAL PARTS from Line 1?3. Open-ended challenge — design a fair-share scenario
This question has more than one valid answer. 4 marks
3.1 Design your OWN ratio-split scenario. Make sure the total divides cleanly into the parts (no decimal shares).
(i) Choose three people (or three groups) to share something. Write down a short scenario sentence.
(ii) Choose a ratio with three parts where the parts sum to a number between 6 and 15 (e.g., 2 : 3 : 4 or 1 : 3 : 5).
(iii) Choose a total amount (money, mass, or time) such that the total ÷ sum of parts is a whole number.
(iv) Work out each share using the unitary method and check that the three shares add back to your total.
Bonus: Re-do the scenario with the SAME ratio but DOUBLE the total. What happens to each share?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $700 in 3 : 4
Parts = 7; 1 part = $100. Shares: $300 and $400. Check: $300 + $400 = $700 ✓.
1.2 — 45 kg in 1 : 2 : 6
Parts = 9; 1 part = 5 kg. Shares: 5 kg, 10 kg, 30 kg. Check: 5 + 10 + 30 = 45 ✓.
1.3 — $360 in 5 : 3 : 2
Parts = 10; 1 part = $36. Shares: $180, $108, $72. Check: 180 + 108 + 72 = $360 ✓.
1.4 — 48 kg in 3 : 4 : 5, middle share
Parts = 12; 1 part = 48 ÷ 12 = 4 kg. Middle share (4 parts) = 4 × 4 = 16 kg.
1.5 — Reverse: 4 : 5, bigger share $200
Bigger share is 5 parts = $200, so 1 part = $40. Total parts = 9. Total = 9 × $40 = $360. (Or: smaller share = 4 × $40 = $160; $160 + $200 = $360.)
1.6 — $540 in 1 : 2 : 3
Parts = 6; 1 part = $90. Smallest = $90, largest = $270, difference = $180. Check: 90 + 180 + 270 = $540 ✓.
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong "1 part" value is then used in Line 3).
(b) The student divided $540 by 3 (the number of people / largest ratio number), but they should have divided by 6 (the TOTAL PARTS from Line 1). The sum check fails: $180 + $360 + $540 = $1080, not $540 — the shares overshoot by double.
(c) Corrected working:
Total parts = 1 + 2 + 3 = 6.
Value of 1 part = $540 ÷ 6 = $90.
Shares: 1 × $90 = $90, 2 × $90 = $180, 3 × $90 = $270.
Sum check: $90 + $180 + $270 = $540 ✓.
Corrected shares: $90, $180, $270.
3 — Open-ended challenge (sample solution)
Sample scenario: "Three students split $80 babysitting money in the ratio 2 : 3 : 3 because two of them stayed the whole night and one left early."
Sum of parts = 2 + 3 + 3 = 8. Total ÷ parts = $80 ÷ 8 = $10 per part.
Shares: 2 × $10 = $20, 3 × $10 = $30, 3 × $10 = $30.
Sum check: 20 + 30 + 30 = $80 ✓.
Bonus: Double the total ($160 in 2 : 3 : 3). 1 part now = $20. Shares: $40, $60, $60 (each share is also doubled). The ratio is unchanged.
Other valid scenarios: $60 in 1 : 2 : 3 ($10, $20, $30); 90 minutes in 2 : 3 : 4 (20, 30, 40 min); 120 kg in 3 : 4 : 5 (30, 40, 50 kg).
Marking: 1 mark for a plausible scenario sentence; 1 mark for a valid ratio with whole-number 1-part value; 1 mark for correct shares with sum check; 1 mark for the doubling bonus answered correctly.