Mathematics • Year 8 • Unit 1 • Lesson 3
Percentage of a Quantity — Mixed Challenge
Pull together every idea from Lessons 1-3: percentage conversion, the multiplier and unitary methods, and building percentages from blocks (10% / 5% / 1%). Six mixed problems, a "find the mistake", and an open-ended challenge.
1. Mixed problems — choose the right method
Each question uses one (or both) of the methods from Lesson 3. Decide which method fits before you start writing. Show your working. 3 marks each
1.1 Find 40% of $250 (use the multiplier method).
1.2 Find 35% of 80 kg using the unitary method (show the 1% step).
1.3 A $120 shirt has 25% off. (a) Find the discount. (b) Find the sale price.
1.4 Find 12.5% of $96 (use any method).
1.5 Find 110% of $50. (Yes, more than 100% is allowed — your answer will be more than $50.)
1.6 Find 28% of $1500 using mental maths (show how you built it from 10%, 5%, and 1% blocks).
2. Find the mistake
Another student has tried to find 30% of $50. 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 — find 30% of $50:
Line 1: "% of" means "multiply by the percent as a decimal".
Line 2: 30% as a decimal = 3.0.
Line 3: 3.0 × 50 = 150.
Line 4: So 30% of $50 = $150.
(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: 30% of $50 should be LESS than $50 (you're taking less than half). $150 is three times the original — that's a giant red flag.3. Open-ended challenge — build that percentage
This question has more than one valid answer. 4 marks
3.1 A friend says they can find 17.5% of any whole-dollar amount in their head using only the building blocks 10%, 5%, 1% (and halves of those).
(i) Show how to build 17.5% from those blocks (which blocks add to 17.5%?).
(ii) Use your method to find 17.5% of $200 in your head.
(iii) Use your method again to find 17.5% of $80.
(iv) Check ONE of your answers using the multiplier method.
Tip: Think of 17.5% as 10% + 5% + 2.5%, where 2.5% is half of 5%.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 40% of $250
0.40 × 250 = $100.
1.2 — 35% of 80 kg (unitary)
1% of 80 = 80 ÷ 100 = 0.8 kg. Then 35 × 0.8 = 28 kg.
1.3 — $120 shirt with 25% off
(a) Discount = 0.25 × 120 = $30.
(b) Sale price = $120 − $30 = $90.
1.4 — 12.5% of $96
12.5% = 0.125. 0.125 × 96 = $12. (Bonus: 12.5% = 1/8, and 96 ÷ 8 = 12. Same answer, faster route.)
1.5 — 110% of $50
110% = 1.10. 1.10 × 50 = $55. (Makes sense: 100% would be $50; the extra 10% adds $5.)
1.6 — 28% of $1500 (mental)
10% of $1500 = $150. So 20% = $300. 5% = half of 10% = $75. 1% = $15, so 3% = $45.
Then 28% = 20% + 5% + 3% = $300 + $75 + $45 = $420.
(Multiplier check: 0.28 × 1500 = $420. ✓)
2 — Find the mistake
(a) The mistake is on Line 2 (then carried into Lines 3 and 4).
(b) 30% does NOT equal 3.0. The student moved the decimal point ONE place instead of TWO. Correct: 30% = 30 ÷ 100 = 0.30.
(c) Corrected working:
"% of" means "multiply by the percent as a decimal".
30% = 30 ÷ 100 = 0.30.
0.30 × 50 = 15.
So 30% of $50 = $15.
Sanity check: 30% should be less than half, so the answer should be less than $25. $15 fits; $150 does not.
3 — Open-ended challenge (sample solution)
(i) Build 17.5% as 10% + 5% + 2.5% (where 2.5% is half of 5%).
(ii) 17.5% of $200: 10% = $20, 5% = $10, 2.5% = $5. Total = $35.
(iii) 17.5% of $80: 10% = $8, 5% = $4, 2.5% = $2. Total = $14.
(iv) Multiplier check on (ii): 0.175 × 200 = $35. ✓
Marking: 1 mark for the correct breakdown (10 + 5 + 2.5); 1 mark for each of the two mental answers; 1 mark for the multiplier check. Other valid breakdowns (e.g. 15% + 2.5%) also earn full marks if the working is correct.