Mathematics • Year 8 • Unit 1 • Lesson 6
Percentage Decrease — Mixed Challenge
Pull together everything from Lesson 6: subtract method, multiplier method, depreciation and chained discounts. 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 ideas from Lesson 6. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 Decrease $560 by 25%.
1.2 A $480 bicycle has 18% off. Find the sale price.
1.3 If you save $24 on a $96 item, what percentage discount did you receive?
1.4 A $1200 phone is reduced by 12.5%. Find the sale price.
1.5 A delivery van worth $50 000 loses 22% in year 1 and then a further 18% in year 2. What is it worth at the end of year 2? (Apply the multipliers in sequence — do NOT add the percentages.)
1.6 A $90 backpack is "40% off". What is the sale price, AND what is the discount in dollars?
2. Find the mistake
A Year 8 student tried to find the sale price of a $250 jacket with 30% off. 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 — sale price of $250 jacket at 30% off:
Line 1: "30% off" means I save 30%, so I need a multiplier.
Line 2: Multiplier = 1 + 0.30 = 1.30 (because we want what's left after taking 30%).
Line 3: Sale = 250 × 1.30
Line 4: Sale = $325. (The shopper pays MORE than the original!)
(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 sale price.
Stuck? A DECREASE makes the price SMALLER. So the multiplier must be LESS than 1. Sanity check: 30% off must give you less than $250, not more.3. Open-ended challenge — sale-tag designer
This question has more than one valid answer. 4 marks
3.1 A shop owner wants to advertise a jacket — originally $200 — with a sale price of exactly $140. They are stuck on how to word the sign.
(i) Find the single "X% off" sale that would give a final price of $140.
(ii) Find two different ways to write the sale as a CHAINED discount (e.g. "Y% off, then a further Z% at the till"). Both must end at exactly $140. Show working for each.
(iii) Of your three options (one single and two chained), which sounds the MOST appealing to a shopper? Briefly justify.
Hint for (ii): Pick a Y, then work out what Z makes the second multiplier give $140.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Decrease $560 by 25%
Multiplier = 0.75. New = 560 × 0.75 = $420.
1.2 — $480 bicycle, 18% off
Multiplier = 0.82. Sale = 480 × 0.82 = $393.60.
1.3 — Save $24 on a $96 item
% off = (24 / 96) × 100 = 25%.
1.4 — $1200 phone, 12.5% off
Multiplier = 1 − 0.125 = 0.875. Sale = 1200 × 0.875 = $1050.
1.5 — $50 000 van, 22% then 18% depreciation
Year 1: 50 000 × 0.78 = $39 000. Year 2: 39 000 × 0.82 = $31 980.
(Adding 22% + 18% = 40% would give $30 000, which is wrong by $1980 — the second loss acts on the smaller post-year-1 value.)
1.6 — $90 backpack, 40% off
Discount = 0.40 × 90 = $36. Sale = 90 − 36 = $54 (or 90 × 0.60 = $54).
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong multiplier is carried into Lines 3 and 4).
(b) The student wrote 1 + 0.30 = 1.30, which would INCREASE the price by 30% rather than decreasing it. A decrease multiplier must be LESS than 1. The correct formula is 1 − 0.30 = 0.70.
(c) Corrected working:
"30% off" means I save 30%, so I pay 70%.
Multiplier = 1 − 0.30 = 0.70.
Sale = 250 × 0.70 = $175.
Sanity check: a sale price should be LESS than $250, not more. ✓
3 — Open-ended challenge (sample solution)
We need to get from $200 to $140 ($60 off).
(i) Single discount: 200 × m = 140 ⇒ m = 0.70 ⇒ "30% off".
(ii) Two chained options:
Option A — "20% off, then a further 12.5% at the till":
200 × 0.80 = $160. Then 160 × 0.875 = $140. ✓ (Z = 12.5%)
Option B — "10% off, then a further 22.22% at the till" (use ~22%):
200 × 0.90 = $180. Then need 180 × m = 140, so m = 140/180 ≈ 0.7778, meaning 22.22% off. ✓ (Slightly messier numbers but valid.)
Other valid pairs: 25% then 6.67%, 15% then 17.6%, 40% then "−16.67%" (which would be an increase — invalid for a sale!), etc.
(iii) Either of the chained options often sounds MORE appealing because shoppers see two reductions and feel they are getting a bigger deal. However, all three give exactly $140. Marketing departments routinely use the "chained" wording because the same final price feels like a better bargain.
Marking: 1 mark for the single 30%-off answer; 2 marks for two different valid chained options with correct working; 1 mark for a sensible justification in (iii).