Mathematics • Year 8 • Unit 1 • Lesson 19
Multi-Step Financial — Mixed Challenge
Pull together everything from Lesson 19: markups, discounts, GST, ratio splits and combined multipliers. 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 multipliers. Decide which moves apply before you start writing. Use multipliers and show your working. 3 marks each
1.1 A $100 item gets +20% markup, then −20% sale, then +10% GST. Find the final price.
1.2 A $132 inc-GST item gets a 10% loyalty discount. Final price.
1.3 A $50 cost item gets +50% markup, then 20% off. Find the sale price.
1.4 Split $420 in the ratio 2 : 3 : 1. Find the second person's share.
1.5 A car worth $30 000 depreciates 20%, then another 15%. Final value.
1.6 A $300 inc-GST restaurant bill is split 3 ways equally. (a) GST in the bill? (b) Each person pays?
2. Find the mistake
Another student has tried to solve "A $100 jacket is marked up 25% then discounted 25%. Find the final price." 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: +25% markup multiplier is 1.25.
Line 2: −25% discount multiplier is 0.75.
Line 3: Since +25 and −25 cancel out, final price = original = $100.
Line 4: Final answer: $100.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: try the actual calculation $100 × 1.25 × 0.75.)
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Lines 1 and 2 are correct (the multipliers ARE right). The fault is in the assumption made in Line 3. Multipliers must be MULTIPLIED, not added/subtracted.3. Open-ended challenge — design a retail markup / sale chain
This question has more than one valid answer. 4 marks
3.1 You run a small online shop. You buy a $40 cost-price item. You want the customer to eventually pay around $60 inc-GST after a markup, then a sale discount, then GST is added.
(i) Choose a markup percentage (between 50% and 100%) and write its multiplier.
(ii) Choose a discount percentage (between 10% and 40%) and write its multiplier.
(iii) Apply both multipliers and then +10% GST (× 1.10) to the $40 cost.
(iv) Check that your final price is reasonably close to $60 — if it's not, adjust your markup or discount and try again.
Bonus: What is your profit per item in dollars? (Profit = final customer price − $40 cost.)
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $100, +20% / −20% / +10% GST
$100 × 1.20 × 0.80 × 1.10 = $100 × 1.056 = $105.60.
1.2 — $132 inc-GST, 10% loyalty off
$132 × 0.90 = $118.80.
1.3 — $50, +50% then −20%
$50 × 1.50 × 0.80 = $50 × 1.20 = $60.
1.4 — $420 in 2 : 3 : 1, second share
Total parts = 6. 1 part = $420 ÷ 6 = $70. Second share (3 parts) = 3 × $70 = $210.
1.5 — $30 000 car, −20% then −15%
$30 000 × 0.80 × 0.85 = $30 000 × 0.68 = $20 400.
1.6 — $300 inc-GST bill, split 3
(a) GST = $300 ÷ 11 ≈ $27.27.
(b) Each pays = $300 ÷ 3 = $100.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) +25% and −25% don't cancel because the multipliers are 1.25 and 0.75, and 1.25 × 0.75 = 0.9375 (NOT 1). The percentages are taken from different starting amounts, so the discount removes more value than the markup added.
(c) Corrected working:
+25% markup multiplier = 1.25.
−25% discount multiplier = 0.75.
Final = $100 × 1.25 × 0.75 = $100 × 0.9375 = $93.75.
Sanity check: $100 × 1.25 = $125; then $125 × 0.75 = $93.75 ✓.
3 — Open-ended challenge (sample solution)
Sample plan: +60% markup, 23% off, +10% GST.
Multipliers: 1.60, 0.77, 1.10. Combined: 1.60 × 0.77 × 1.10 = 1.3552.
Final price = $40 × 1.3552 ≈ $54.21. (A bit under $60 — could push markup higher.)
Better attempt: +75% markup, 20% off, +10% GST. Multipliers: 1.75, 0.80, 1.10. Combined = 1.54. Final = $40 × 1.54 = $61.60 (close to $60 ✓).
Bonus: Profit per item = $61.60 − $40 = $21.60.
Other valid combos that land near $60: +80%/−25%/+10% GST → $40 × 1.80 × 0.75 × 1.10 = $59.40; +100%/−30%/+10% GST → $40 × 2.00 × 0.70 × 1.10 = $61.60.
Marking: 1 mark for each multiplier correctly identified (up to 2 marks); 1 mark for the final inc-GST customer price; 1 mark for a profit calculation in dollars.