Mathematics • Year 8 • Unit 1 • Lesson 7
Discounts — Mixed Challenge
Pull together everything from Lesson 7: forward (find sale), reverse (find marked), find % off, and layered/chained discounts. Six mixed problems, one "find the mistake" on a reverse calculation, and one open-ended challenge.
1. Mixed problems — choose the right move
Each question tests a different combination of ideas from Lesson 7. Decide whether you are going forward, backward, or computing a percentage before you start writing. Show your working. 3 marks each
1.1 A $320 watch has 40% off. Find the sale price.
1.2 A sale price of $144 reflects 20% off. What was the marked price?
1.3 A $60 book is now $45. What percentage discount has been applied?
1.4 A $450 chair is on sale for $360. (a) Saving in dollars? (b) % off? (c) What pay-fraction was used?
1.5 A store advertises "Up to 50% off!". The fine print: 20% off, then a further 25% at the till, then a final 15% student discount. Find the overall multiplier and the equivalent SINGLE percentage discount.
1.6 A pair of boots on sale for $84 represents a 30% discount. Find the marked price AND the dollar saving.
2. Find the mistake
A Year 8 student tried to find the marked price of a jacket on sale for $60 after a 25% discount. 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 — marked price for sale = $60, 25% off:
Line 1: 25% off means I saved 25%, so pay-fraction = 0.75.
Line 2: To go from sale back to marked, multiply by 0.75 again.
Line 3: Marked = 60 × 0.75
Line 4: Marked = $45. (Marked is LESS than sale — impossible, but I'll go with it.)
(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 marked price.
Stuck? Sanity check: marked price MUST be larger than sale price. Reverse problems use DIVISION, not multiplication.3. Open-ended challenge — design a sale headline
This question has more than one valid answer. 4 marks
3.1 A shop owner wants every customer to pay exactly $63 for a jacket that has a marked price of $90. They are testing three different sale-tag wordings, but every wording must land the customer at exactly $63.
For each wording, write the headline and show working that confirms it gives a final price of $63:
(i) A SINGLE "X% off" deal.
(ii) A CHAINED deal: "Y% off, then a further Z% off at the till" — choose any sensible Y between 5% and 25%, then work out the matching Z.
(iii) A "save $A" deal — work out A and confirm the final price.
Bonus: Of your three wordings, which do you think a typical Year 8 shopper would find most appealing? Justify briefly.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $320 watch, 40% off
Sale = 320 × 0.60 = $192.
1.2 — Sale $144, 20% off
Marked = 144 ÷ 0.80 = $180.
1.3 — $60 book now $45
Saving = $15. % off = (15 / 60) × 100 = 25% off.
1.4 — $450 chair on sale for $360
(a) Saving = 450 − 360 = $90.
(b) % off = (90 / 450) × 100 = 20% off.
(c) Pay-fraction = 360 / 450 = 0.80 (since 1 − 0.20 = 0.80).
1.5 — "Up to 50% off" chain (20% then 25% then 15%)
Combined multiplier = 0.80 × 0.75 × 0.85 = 0.510. Equivalent single discount = 1 − 0.510 = 49%. Just under the headline 50% — technically "up to" so the ad is borderline acceptable.
1.6 — Sale $84 after 30% off
Pay-fraction = 0.70. Marked = 84 ÷ 0.70 = $120. Saving = 120 − 84 = $36.
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong operation is then carried into Lines 3 and 4).
(b) To go BACKWARDS from the sale price to the marked price, you must DIVIDE by the pay-fraction, not multiply. Multiplying by 0.75 a second time shrinks the price further, but the marked price should be LARGER than the sale price.
(c) Corrected working:
25% off means pay-fraction = 0.75.
Marked = Sale ÷ pay-fraction = 60 ÷ 0.75 = $80.
Check: 80 × 0.75 = $60 ✓. The shopper saved $20.
3 — Open-ended challenge (sample solution)
We need to land at $63 from $90 (a saving of $27).
(i) Single discount: 90 × m = 63 ⇒ m = 0.70 ⇒ "30% off".
(ii) Chained option (Y = 10%): 90 × 0.90 = $81. Then 81 × (1 − Z/100) = 63 ⇒ (1 − Z/100) = 63/81 ≈ 0.7778 ⇒ Z ≈ 22.2%. Headline: "10% off, plus a further 22.2% at the till".
Other valid Ys (with matching Z): Y = 15% ⇒ Z ≈ 17.6%; Y = 20% ⇒ Z = 12.5%; Y = 25% ⇒ Z ≈ 6.67%.
(iii) Save-$ wording: Save = 90 − 63 = $27. Headline: "Save $27 on every jacket!". Final price = 90 − 27 = $63 ✓.
Bonus: Sample answer — A typical shopper might find the chained option most appealing because two reductions sound like a bigger deal, even though all three give the same $63. Save-$ headlines work well when the dollar amount is large and concrete.
Marking: 1 mark for the single 30%-off answer; 1 mark for a valid chained (Y, Z) with working; 1 mark for the "save $27" wording with check; 1 mark for a sensible bonus justification.