Mathematics • Year 8 • Unit 1 • Lesson 5
Percentage Increase — Mixed Challenge
Pull together every idea from Lessons 1-5: converting between FDP, finding a percentage of a quantity, expressing a quantity as a percentage, and applying a percentage increase via the add or multiplier methods.
1. Mixed problems — choose the right move
Each question uses one (or more) of the unit's big ideas. Decide what's being asked before you write. Show your working. 3 marks each
1.1 Increase $480 by 15% using the multiplier method.
1.2 A $90 price tag rises to $99. (a) What is the dollar increase? (b) What is the percentage increase? (Hint: increase as a % of the ORIGINAL.)
1.3 A movie ticket is $18. It is increased by 12.5%. Find the new price.
1.4 A book's GST-inclusive price is $33. The GST is 10% (added on top of the GST-free price). Find the GST-free price. (Hint: $33 = GST-free × 1.10.)
1.5 A car's value drops from $24 000 to $22 800 in a year. (a) What is the dollar DECREASE? (b) What is the percentage decrease? (Hint: same idea as percentage increase, just subtract.)
1.6 A $250 jacket is increased by 20%, then the new price is increased by another 10%. (a) Find the price after BOTH increases. (b) Is this the same as a single 30% increase on $250? Show your reasoning.
2. Find the mistake
A student has tried to increase $60 by 25% using the multiplier method. 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 — increase $60 by 25%:
Line 1: Formula: new = original × (1 + P/100).
Line 2: P = 25, so multiplier = 1 + 25/100 = 1.25.
Line 3: new = 60 × 0.25 = 15.
Line 4: So the new price is $15.
(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 final answer.
Stuck? Sanity check: an INCREASE should give a price BIGGER than $60, not smaller. $15 is the discount-style answer — the wrong direction entirely.3. Open-ended challenge — design the rises
This question has more than one valid answer. 4 marks
3.1 A worker's pay starts at $20/hour. After two annual pay rises (one rise this year, another rise next year), their pay is $23.04/hour.
Find three different pairs of percentage rises that get from $20 to $23.04 in exactly two steps.
For each pair:
(i) State the two percentage rises (call them P₁% and P₂%).
(ii) Show the two multiplications: $20 × multiplier₁ × multiplier₂ = $23.04.
(iii) Note whether the SAME percentage rise applied twice would also work (and what it would be).
Tip: The combined multiplier must equal 23.04 ÷ 20 = 1.152. Any two multipliers whose product is 1.152 will work.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $480 + 15%
$480 × 1.15 = $552.
1.2 — $90 → $99
(a) Increase = $99 − $90 = $9.
(b) Percentage increase = (9 ÷ 90) × 100 = 10 × 1 = 10% (using the L4 formula: increase as a % of the original).
1.3 — $18 + 12.5%
$18 × 1.125 = $20.25. (Add method check: 12.5% of $18 = $2.25; $18 + $2.25 = $20.25. ✓)
1.4 — Reverse GST
GST-free price × 1.10 = $33, so GST-free price = $33 ÷ 1.10 = $30. (Check: 10% of $30 = $3; $30 + $3 = $33. ✓)
1.5 — Car value drop
(a) Decrease = $24 000 − $22 800 = $1200.
(b) Percentage decrease = (1200 ÷ 24 000) × 100 = 0.05 × 100 = 5%.
1.6 — Two stacked increases
(a) $250 × 1.20 = $300, then $300 × 1.10 = $330.
(b) A single 30% increase would give $250 × 1.30 = $325. So NO — two separate increases (20% then 10%) give $330, NOT the same as one 30% increase. The two methods give different answers because the second 10% is applied to the BIGGER $300, not the original $250. (Combined multiplier: 1.20 × 1.10 = 1.32, equivalent to a single 32% increase — not 30%.)
2 — Find the mistake
(a) The mistake is on Line 3 (then Line 4 is wrong as a result).
(b) The student multiplied by 0.25 instead of by the multiplier 1.25 from Line 2. 0.25 × 60 = 15 gives the INCREASE only, not the new total. They need to use the multiplier they correctly calculated.
(c) Corrected working:
Formula: new = original × (1 + P/100).
P = 25, so multiplier = 1 + 25/100 = 1.25.
new = 60 × 1.25 = 75.
So the new price is $75.
Sanity check: an increase makes the price bigger; $75 > $60 — that's the right direction.
3 — Open-ended challenge (sample solution)
We need two multipliers whose product is 23.04 ÷ 20 = 1.152.
Pair 1: 20% rise then a 4% DROP (multipliers 1.20 and 0.96). Check: $20 × 1.20 × 0.96 = $24 × 0.96 = $23.04. ✓ (Year 2 is technically a decrease, but the question allows any P₁ and P₂ — including negative.)
Pair 2: 8% rise then another 6.67% rise (multipliers 1.08 and ≈1.0667). Check: 1.08 × 1.0667 ≈ 1.152, so $20 × 1.152 = $23.04. ✓
Pair 3: 15% rise then a 0.17% rise (multipliers 1.15 and ≈1.00174). Check: 1.15 × 1.00174 ≈ 1.152. So $20 × 1.152 = $23.04. ✓
Same percentage twice: if the SAME multiplier m is used both years, then m × m = 1.152, so m = √1.152 ≈ 1.0733. That's a 7.33% rise each year.
Marking: 1 mark for each valid pair with working (up to 3). 1 mark for the "same percentage twice" answer using the square root (≈ 7.33%). Other valid pairs with combined multiplier 1.152 also earn full marks.