Mathematics • Year 7 • Unit 1 • Lesson 15
Unit 1 in the Real World
Use Unit 1 skills together: pricing and discounts, sharing in a ratio, scaling a recipe, comparing best value. Each problem combines two or three Unit 1 topics in a single real scenario.
1. Word problems
Each problem combines two or three Unit 1 topics. Read carefully, identify which methods you need, and show every step. A final answer with no working only earns half marks.
1.1 — Shop markup and sale (percentages × 2). A shop buys an item for $80 and marks it up by 35% to set the selling price.
(a) What is the selling price?
(b) In a later sale, the shop offers 20% off the selling price. What is the sale price?
(c) What is the overall percentage profit, calculated from the original $80 cost to the sale price? 3 marks
1.2 — Recipe in a ratio (fractions + ratios). A recipe uses flour and sugar in the ratio 5:2. You decide to use 3/4 kg of flour.
(a) How much sugar do you need? Show the ratio working.
(b) Express your answer to (a) as a decimal (in kg).
(c) What is the total weight of flour + sugar? 3 marks
1.3 — Group dinner (decimals + ratios). Six friends go out for dinner. The total bill (food + drinks) is $174.60. They agree to split the bill in the ratio 1:1:1:1:1:1 — that is, evenly.
(a) How much does each person pay?
(b) After the meal they realise that one person, Lin, had only a $12.50 main and no drinks. The other five agree to absorb the difference between Lin's bill and the equal split. How much extra does each of the five other friends pay? 3 marks
1.4 — Pre-GST and after-GST (percentages). A laptop is advertised at $1,320 including GST (GST is 10%).
(a) What was the price before GST was added? (Hint: divide by 1.10, not subtract 10%.)
(b) Confirm your answer by adding 10% GST back to the pre-GST price and checking you get $1,320. 2 marks
1.5 — Smarter shopping (best value across formats). A supermarket sells the same brand of laundry detergent in three sizes:
- 750 mL for $7.50
- 1.5 L for $13.20
- 3 L for $24.00
(a) Find the price per litre for each pack.
(b) Which size is the best value, and which is the worst?
(c) The household uses 0.25 L per week. Using the best-value pack, how many weeks of supply does $50 buy you? (Whole weeks only.) 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says "If a $100 shirt is increased by 10% and then decreased by 10%, you end up back at $100 — because the 10% adds and then takes away." In your own words explain (i) whether the classmate is correct, (ii) show the two calculations with working, (iii) state the final price, and (iv) explain in one sentence why the answer is not exactly $100.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Shop markup and sale
(a) Selling price = $80 × 1.35 = $108.
(b) Sale price = $108 × 0.80 = $86.40.
(c) Profit = $86.40 − $80 = $6.40. Profit % = $6.40 ÷ $80 × 100 = 8%.
1.2 — Recipe in a ratio
(a) Ratio 5:2 with 3/4 kg of flour. One ratio part = (3/4) ÷ 5 = 3/20 kg. Sugar = 2 × 3/20 = 6/20 = 3/10 kg.
(b) 3/10 kg = 0.3 kg.
(c) Total = 3/4 + 3/10 = 15/20 + 6/20 = 21/20 kg = 1.05 kg.
1.3 — Group dinner
(a) Equal split: $174.60 ÷ 6 = $29.10 each.
(b) Lin pays $12.50. Difference to absorb = $29.10 − $12.50 = $16.60. Spread over 5 friends: $16.60 ÷ 5 = $3.32 extra each (so each pays $29.10 + $3.32 = $32.42).
1.4 — Pre-GST and after-GST
(a) Pre-GST = $1,320 ÷ 1.10 = $1,200.
(b) Check: $1,200 × 1.10 = $1,320 ✓.
1.5 — Best value detergent
(a) 750 mL = 0.75 L → $7.50 ÷ 0.75 = $10.00/L.
1.5 L → $13.20 ÷ 1.5 = $8.80/L.
3 L → $24.00 ÷ 3 = $8.00/L.
(b) Best: 3 L pack ($8.00/L). Worst: 750 mL pack ($10.00/L).
(c) Buying 3 L packs at $8.00/L: $50 buys 50 ÷ 8 = 6.25 L. At 0.25 L per week, supply = 6.25 ÷ 0.25 = 25 weeks.
2.1 — Explain your thinking (sample response)
The classmate is wrong — even though it sounds like the 10% should cancel out, you don't end up at $100.
Step 1: $100 × 1.10 = $110 (after the 10% increase).
Step 2: $110 × 0.90 = $99 (after the 10% decrease, applied to the new $110 price).
Final price = $99, which is $1 less than the original.
It's not exactly $100 because the second 10% is taken off the new larger amount ($110), so the dollar amount removed ($11) is bigger than the dollar amount added ($10). Percentage increases and decreases of the same size never quite cancel — there is always a small loss.
Marking: 1 for saying "wrong"; 1 for the $110 step; 1 for the $99 step; 1 for explaining that the 10% applies to a different base each time.