Mathematics • Year 8 • Unit 1 • Lesson 20
Unit 1 Synthesis — Mixed Challenge
A final mixed challenge across the whole of Unit 1: FDP, percentage of, profit/loss, rates, ratios, GST and multi-step price changes. 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 unit ideas. Decide which moves apply before you start writing. Show all working. 3 marks each
1.1 Convert 7/8 to a percentage. Then find 7/8 of $240.
1.2 Find 18% of $240.
1.3 Split $315 in the ratio 4 : 5.
1.4 A car bought for $20 000 is sold for $24 000. Find: (a) profit in $, (b) % profit on cost.
1.5 A jacket has cost $80, marked up 40%, then a 15% till discount. Find (a) the selling price, (b) the profit per jacket.
1.6 A 1.5 kg bag of apples costs $6. (a) What's the unit price per kg? (b) Each apple weighs roughly 150 g. How many apples are in the bag, and what's the price per apple?
2. Find the mistake
Another student has tried to find the GST that's hidden in a $66 inc-GST item, then use it. 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:
Line 1: The item is inc-GST, so GST is 10% of the total.
Line 2: GST = 0.10 × $66 = $6.60.
Line 3: Pre-GST price = $66 − $6.60 = $59.40.
Line 4: Final answer: GST = $6.60, pre-GST price = $59.40.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: try the sanity check — does pre-GST × 1.10 equal $66?)
(c) Write out the corrected working in full, including the corrected GST and pre-GST price.
Stuck? GST is 10% of the PRE-GST price, not 10% of the inc-GST total. The shortcut for inc-GST totals: GST = total ÷ 11 (because $1.10 contains $0.10 of GST, and 0.10/1.10 = 1/11).3. Open-ended challenge — design a small honest business
This question has more than one valid answer. 4 marks
3.1 You want to start a small T-shirt business. You will buy 50 T-shirts wholesale at $8 each (cost price). You want to make at least $300 profit total after selling all 50 shirts, but you also want the retail price to be UNDER $25 so people will actually buy them.
(i) Choose a markup percentage (anywhere from 50% to 150%).
(ii) Work out the retail price per shirt (= $8 × markup multiplier).
(iii) Calculate total revenue if you sell all 50, and total profit (revenue − $400 total cost).
(iv) Check both conditions: profit ≥ $300 AND retail price < $25. Adjust your markup and try again if either fails.
Bonus: Add 10% GST to your retail price. What's the customer-facing inc-GST price? And what's the GST you owe the government per shirt?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 7/8 of $240
7/8 = 7 ÷ 8 = 0.875 = 87.5%. Then 7/8 of $240 = $240 × 7/8 = $30 × 7 = $210.
1.2 — 18% of $240
0.18 × $240 = $43.20. (Or: 10% = $24; 5% = $12; 1% = $2.40; 3% = $7.20; sum 18% = 24 + 12 + 7.20 = $43.20.)
1.3 — $315 in 4 : 5
Parts = 9; 1 part = $35. Shares: $140 and $175.
1.4 — Car $20 000 → $24 000
(a) Profit = $24 000 − $20 000 = $4000.
(b) % profit on cost = 4000/20 000 × 100 = 20%.
1.5 — Jacket $80, +40% then −15%
(a) Selling price = $80 × 1.40 × 0.85 = $80 × 1.19 = $95.20.
(b) Profit = $95.20 − $80 = $15.20.
1.6 — Apples 1.5 kg for $6
(a) Unit price = $6 ÷ 1.5 = $4/kg.
(b) 1.5 kg = 1500 g. 1500 ÷ 150 = 10 apples. Price per apple = $6 ÷ 10 = $0.60.
2 — Find the mistake
(a) The mistake is on Line 1 (and it leads to a wrong GST in Line 2 and a wrong pre-GST price in Line 3).
(b) GST is 10% of the PRE-GST price, not 10% of the inc-GST total. So "GST = 0.10 × $66" is wrong. Sanity check: if pre-GST is $59.40, then × 1.10 = $65.34, NOT $66 — so something is off.
(c) Corrected working:
For inc-GST totals, GST = total ÷ 11 = $66 ÷ 11 = $6.
Pre-GST price = $66 − $6 = $60.
Sanity check: $60 × 1.10 = $66 ✓.
3 — Open-ended challenge (sample solution)
Sample plan: +150% markup. Multiplier = 2.50. Retail = $8 × 2.50 = $20 per shirt.
Total revenue (all 50 sold) = 50 × $20 = $1000. Total cost = 50 × $8 = $400.
Total profit = $1000 − $400 = $600. Check both conditions: $600 ≥ $300 ✓ AND $20 < $25 ✓. Plan works.
Bonus: Inc-GST customer price = $20 × 1.10 = $22. GST per shirt = $22 − $20 = $2 (or $22 ÷ 11 = $2).
Other valid markups: +100% → retail $16 (profit only $400, just over the $300 mark, works); +125% → retail $18 (profit $500 ✓); +180% → retail $22.40 (profit $720, but customer price after GST is $24.64, still under $25 ✓).
Marking: 1 mark for a markup that gives retail under $25; 1 mark for correct total revenue and profit; 1 mark for showing BOTH conditions are met (profit ≥ $300 AND retail < $25); 1 mark for the bonus inc-GST price + GST per shirt.