Mathematics • Year 8 • Unit 1 • Lesson 4
Quantity as a Percentage — Mixed Challenge
Pull together every idea from Lessons 1-4: turn a fraction into a decimal and a percentage, find a percentage of a quantity, and express one quantity as a percentage of another. Six mixed problems, a "find the mistake", and an open-ended challenge.
1. Mixed problems — choose the right move
Each question uses one (or more) of the four big moves from this unit so far. Decide what's being asked before you write. Show your working. 3 marks each
1.1 Express 24 out of 32 as a percentage.
1.2 Express 750 mL as a percentage of 2 L. (Hint: convert to the same units.)
1.3 A teacher scales test marks by adding 5 marks to every score. If Asha originally scored 47 out of 60, her new mark is 52 out of 65 (the maximum also goes up by 5). (a) Find her old percentage. (b) Find her new percentage. (c) Did the scaling actually change her percentage? Explain in one sentence.
1.4 A profit margin is found as (profit ÷ revenue) × 100%. A café made $480 profit on $1500 revenue. What is the profit margin?
1.5 A test is out of 80. To pass you need at least 60%. What is the minimum number of marks you need to pass? (This combines L3 "find a % of a quantity" with L4 thinking.)
1.6 In a Year 8 cohort of 200 students, 78 chose Visual Art as an elective. (a) What percentage chose Visual Art? (b) If next year's cohort is 250 students and the same percentage choose Visual Art, how many students will that be?
2. Find the mistake
A student has tried to express 50 cm as a percentage of 2 m. 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 — express 50 cm as a percentage of 2 m:
Line 1: Use the formula A as % of B = (A ÷ B) × 100%.
Line 2: A = 50, B = 2.
Line 3: (50 ÷ 2) × 100 = 25 × 100 = 2500.
Line 4: So 50 cm is 2500% of 2 m.
(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: 50 cm is half of 1 m, which is a quarter of 2 m. So the answer should be 25%, not 2500%. The error is about units.3. Open-ended challenge — design your own test
This question has more than one valid answer. 4 marks
3.1 Design a small "fair-comparison" test set. Find three different score-out-of-totals that all give exactly 80% as a percentage.
For each score:
(i) Write it in "X out of Y" form.
(ii) Show the working (X ÷ Y) × 100 to confirm it equals 80%.
(iii) Briefly say which "X out of Y" feels easiest to compute mentally and why.
Bonus: Your three totals (Y values) must all be different — no using "4 out of 5", "8 out of 10", "12 out of 15" type chains where one is a simple multiple of another only by 2 ... vary the totals more if you can.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 24 out of 32
(24 ÷ 32) × 100 = 0.75 × 100 = 75%. (Quick check: 24/32 simplifies to 3/4 = 75%.)
1.2 — 750 mL out of 2 L
Convert: 2 L = 2000 mL. Then (750 ÷ 2000) × 100 = 0.375 × 100 = 37.5%.
1.3 — Scaling Asha's mark
(a) Old: 47/60 × 100 ≈ 78.3%.
(b) New: 52/65 × 100 = 0.8 × 100 = 80%.
(c) Yes — the scaling DID change her percentage (from 78.3% to 80%). Adding the same number of marks to top and bottom of a fraction does NOT keep the percentage the same — it actually moves it closer to 100% when both are increased by the same amount.
1.4 — Profit margin
(480 ÷ 1500) × 100 = 0.32 × 100 = 32% profit margin.
1.5 — Pass mark
60% of 80 = 0.60 × 80 = 48 marks needed to pass.
1.6 — Visual Art elective
(a) 78/200 × 100 = 0.39 × 100 = 39%.
(b) 39% of 250 = 0.39 × 250 = 97.5 students. Since you can't have half a student, this would round to about 98 students in practice.
2 — Find the mistake
(a) The mistake is on Line 2 (then the wrong B value is carried through).
(b) The student forgot to convert to the same units. A is in cm (50) but B is in m (2). Either convert B to cm (2 m = 200 cm) or convert A to m (50 cm = 0.5 m). The lesson says "same units, same direction".
(c) Corrected working:
Use the formula A as % of B = (A ÷ B) × 100%.
Convert to the same units: 2 m = 200 cm.
A = 50, B = 200.
(50 ÷ 200) × 100 = 0.25 × 100 = 25.
So 50 cm is 25% of 2 m. ✓
Sanity check: 50 cm is a quarter of 200 cm — that's 25%, which matches.
3 — Open-ended challenge (sample solution)
For 80%, the part must be 4/5 of the whole.
Score 1: 20 out of 25. (20 ÷ 25) × 100 = 80%. ✓ Easy mentally: 25 × 4 = 100, so each mark is worth 4%, and 20 × 4 = 80.
Score 2: 40 out of 50. (40 ÷ 50) × 100 = 80%. ✓ Easy: 50 × 2 = 100, so each mark is 2%, and 40 × 2 = 80.
Score 3: 64 out of 80. (64 ÷ 80) × 100 = 80%. ✓ Slightly harder mentally — but 64/80 simplifies to 4/5 = 80%.
Other valid answers: any X/Y where X = 0.8 × Y, e.g. 12/15, 16/20, 24/30, 32/40, 48/60, 56/70 ...
Marking: 1 mark for each valid score with working (up to 3). 1 bonus mark for varying the totals more than trivially (e.g. picking 25, 50 AND 80 rather than 10, 20, 40 in a simple doubling chain).