Mathematics • Year 8 • Unit 1 • Lesson 12
Rates — Mixed Challenge
Pull together everything from Lesson 12: recognising rates, finding unit rates, scaling, and comparing two rates fairly. Six mixed problems, one “find the mistake”, and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 12. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 A printer prints 120 pages in 5 minutes. Find the unit rate in pages per minute, then use it to find how many pages print in 12 minutes.
1.2 A 750 mL bottle of juice contains 9 g of sugar. Express the sugar content as g per 100 mL.
1.3 A tap fills a 240 L tank in 8 minutes. Find the flow rate in L/min, then find how long it takes to fill a 600 L tank at the same rate.
1.4 A train covers 360 km in 4 hours; a car covers 280 km in 3.5 hours. Which has the higher average speed in km/h, and by how much?
1.5 A wage of $\$22.50$/hour for 7.5 hours of work earns how much? Show your working with the unit rate.
1.6 A car uses 6.4 L per 100 km. (a) How much fuel does it need for 250 km? (b) At $\$1.90$/L, what is the fuel cost for that 250 km trip?
2. Find the mistake
Another student is trying to find the unit price of a 1.5 kg bag of pasta that costs $\$4.50$. 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 the price per kg of pasta:
Line 1: Total cost = $\$4.50$, total weight = 1.5 kg.
Line 2: Unit price = weight ÷ cost = $1.5 \div 4.50$.
Line 3: $1.5 \div 4.50 \approx 0.33$.
Line 4: So the pasta costs $\$0.33$ per kg.
(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 correct unit price.
Stuck? “Per kg” means “divide by the number of kg”. Sanity-check: 1.5 kg of pasta for 33 cents would be very cheap!3. Open-ended challenge — design a fair comparison
This question has more than one valid answer. 4 marks
3.1 A supermarket sells the same brand of yoghurt in three sizes:
- 170 g pot — $\$1.50$
- 500 g tub — $\$3.80$
- 1 kg tub — $\$8.50$
For each size:
(i) Find the unit rate in $/100 g (round to the nearest cent).
(ii) Find the unit rate in $/kg.
(iii) Rank the three sizes from cheapest to most expensive per gram.
(iv) The supermarket prints “family value!” on the 1 kg tub. Is that label fair? Explain in one sentence using your numbers.
Bonus: A customer needs exactly 500 g. List two valid ways to buy 500 g and the total cost of each, then say which is cheapest.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Printer rate
Unit rate = $120 \div 5 = \textbf{24 pages/min}$. In 12 min: $12 \times 24 = \textbf{288 pages}$.
1.2 — Sugar content
Per 100 mL: $9 \div 7.5 = \textbf{1.2 g/100 mL}$ (since 750 mL is 7.5 lots of 100 mL).
1.3 — Tap flow
Flow rate = $240 \div 8 = \textbf{30 L/min}$. For 600 L: $600 \div 30 = \textbf{20 min}$.
1.4 — Train vs car
Train: $360 \div 4 = 90$ km/h. Car: $280 \div 3.5 = 80$ km/h. Train is faster by 10 km/h.
1.5 — Wage
$7.5 \times \$22.50 = \textbf{\$168.75}$.
1.6 — Fuel for 250 km
(a) $250 \div 100 = 2.5$, so litres needed = $2.5 \times 6.4 = \textbf{16 L}$.
(b) Cost = $16 \times \$1.90 = \textbf{\$30.40}$.
2 — Find the mistake
(a) The mistake is on Line 2: the student has divided weight by cost instead of cost by weight. (Lines 3 and 4 then carry the error.)
(b) “Per kg” means “cost per 1 kg”, which is cost ÷ weight, not weight ÷ cost. The wrong way round gives kg per dollar, not dollars per kg.
(c) Corrected: Unit price = cost ÷ weight = $\$4.50 \div 1.5 = \textbf{\$3.00/kg}$. Sanity check: 1.5 kg at $\$3.00$/kg = $\$4.50$. ✓
3 — Open-ended challenge (sample solution)
(i) and (ii):
- 170 g for $\$1.50$: per 100 g = $1.50 \times 100 / 170 \approx \textbf{\$0.88/100 g}$; per kg $\approx \textbf{\$8.82/kg}$.
- 500 g for $\$3.80$: per 100 g = $3.80 / 5 = \textbf{\$0.76/100 g}$; per kg = $\textbf{\$7.60/kg}$.
- 1 kg for $\$8.50$: per 100 g = $\textbf{\$0.85/100 g}$; per kg = $\textbf{\$8.50/kg}$.
(iii) Cheapest to most expensive per gram: 500 g tub ($7.60/kg) < 1 kg tub ($8.50/kg) < 170 g pot ($8.82/kg).
(iv) The “family value!” label is misleading — the 1 kg tub is cheaper per kg than the small pot, but the 500 g tub beats it.
Bonus: Two ways to buy 500 g:
Option A: one 500 g tub = $\$3.80$.
Option B: three 170 g pots = $3 \times 1.50 = \$4.50$ (510 g for $\$4.50$).
Option A is cheaper by $\$0.70$.
Marking: 1 mark for correct unit rates of all three sizes (in either unit); 1 mark for correct ranking; 1 mark for a sensible “family value” judgement supported by numbers; 1 bonus mark for the “two ways to buy 500 g” comparison.