Mathematics • Year 7 • Unit 1 • Lesson 13
Percentages — Mixed Challenge
Pull everything from Lesson 13 together: convert between %, fractions and decimals, find a percentage of an amount, express one number as a % of another, work with discounts, and spot a classic ÷100 mistake. Finish with an open-ended puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 13. Decide which method applies before you start. Show working. 2 marks each
1.1 Convert each to the other two forms: (a) 0.6 → fraction and %; (b) 7/25 → decimal and %; (c) 45% → fraction (simplest) and decimal.
1.2 Find 35% of $240. Show both the decimal method and the mental 10% method.
1.3 Express 9 out of 30 as a percentage. Simplify your fraction first.
1.4 Find 12.5% of 240. (Hint: 12.5% = 1/8.)
1.5 A $50 textbook is reduced by 20%. What is the sale price? Use the multiplier (80%) method.
1.6 A pizza has 8 slices. You eat 3 slices. (a) What fraction of the pizza did you eat? (b) What percentage? (c) What percentage is left?
2. Find the mistake
Another Year 7 student tried to find 15% of 200. Their working is below. Exactly one line contains the mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find 15% of 200:
Line 1: "Of" means multiply, so I need 15% × 200.
Line 2: Convert 15% to a decimal: 15 ÷ 10 = 1.5.
Line 3: Multiply: 1.5 × 200 = 300.
Line 4: Final answer: 15% of 200 = 300.
(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? Per cent means per hundred. Dividing 15 by 10 gives 1.5, but 15% should be much less than 1 — it's less than a fifth. Sanity check the final answer: can 15% of something really be bigger than the original?3. Open-ended challenge — design a sale
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 A shop sells a backpack for $80. The manager wants to advertise a sale where the final price ends in exactly $0.00 (a "whole-dollar" price) and the discount percentage is a whole number between 5% and 50%.
(i) Pick a discount % that satisfies the rules, write down the sale price, and show your working.
(ii) Explain in one or two sentences why your chosen percentage gives a whole-dollar price.
Bonus: How many different whole-number percentages between 5% and 50% give a whole-dollar sale price on this $80 backpack? List them all.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Convert
(a) 0.6 = 6/10 = 3/5, and 0.6 × 100 = 60%.
(b) 7/25 = 28/100 = 0.28, and 7/25 × 100 = 700 ÷ 25 = 28%.
(c) 45% = 45/100 = 9/20 (÷ 5), and 45 ÷ 100 = 0.45.
1.2 — 35% of $240
Decimal: 0.35 × 240 = $84.
Mental: 10% of $240 = $24. 30% = 3 × $24 = $72. 5% = half of 10% = $12. 35% = $72 + $12 = $84 ✓.
1.3 — 9 out of 30 as a %
9/30 = 3/10 (÷ 3). 3/10 × 100 = 30. Answer: 30%.
1.4 — 12.5% of 240
12.5% = 1/8. 240 ÷ 8 = 30. (Or: 0.125 × 240 = 30.)
1.5 — $50 textbook, 20% off
Pay 80%. 0.80 × $50 = $40.
1.6 — Pizza
(a) Eaten = 3/8.
(b) 3/8 × 100 = 300 ÷ 8 = 37.5% eaten.
(c) Left = 100% − 37.5% = 62.5% (or 5/8 of the pizza).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student divided by 10 instead of by 100. Per cent means per hundred, so 15% = 15 ÷ 100 = 0.15, not 1.5.
(c) Corrected working:
Line 1 (kept): 15% × 200.
Line 2 (fixed): 15% = 15 ÷ 100 = 0.15.
Line 3 (fixed): 0.15 × 200 = 30.
Line 4: 15% of 200 = 30.
Sanity check: 15% is a small slice (less than 1/5), so the answer should be much less than 200. 30 is reasonable; 300 is impossible.
3 — Sale design puzzle (sample solution)
Try 25% off: saving = 0.25 × $80 = $20. Sale price = $80 − $20 = $60 ✓ whole dollar, ✓ between 5% and 50%.
Why it works: 25% of $80 = 1/4 of $80 = $20 exactly, with no cents. Any percentage whose decimal form gives a whole number of dollars when multiplied by 80 will work.
Bonus — all valid percentages. Saving = $(80p/100) = $(4p/5). For this to be a whole-dollar amount, p must be a multiple of 5. Whole-number multiples of 5 strictly between 5% and 50% (rules say between, take inclusive): 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. That's 10 valid percentages (with prices $76, $72, $68, $64, $60, $56, $52, $48, $44, $40 respectively).
Marking: 2 marks for any valid % with correct sale price; 1 mark for explaining why it gives a whole-dollar price; 1 bonus mark for the systematic list of all valid percentages (accept any complete reasoning about multiples of 5).