Mathematics • Year 8 • Unit 1 • Lesson 18
Ratio in Context — Mixed Challenge
Pull together everything from Lesson 18: map scales, recipe scaling, mixing solutions, and reverse problems (real → plan). Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different context. Decide which move applies before you start writing. Show your working. 3 marks each
1.1 A map has scale 1 : 200. A 5 cm line on the plan represents what real distance? (Give your answer in metres.)
1.2 A recipe for 4 people uses 200 g of sugar. How much sugar is needed for 6 people, same ratio?
1.3 A drink mix is 1 part juice to 4 parts water. How much juice and water for a 500 mL drink?
1.4 A photo is enlarged in the ratio 2 : 5. A 6 cm side becomes how long?
1.5 A house plan has scale 1 : 100. (a) A real wall is 7 m long. How long is it drawn on the plan? (b) A different feature is 3.5 cm on the plan. How long in real life (m)?
1.6 A nurse mixes a saline solution in the ratio 9 : 1 (water : salt). She only has 30 mL of salt left. How much water should she use to keep the ratio?
2. Find the mistake
Another student has tried to solve "Map scale 1 : 50 000. Two towns are 4.5 cm apart on the map. How far apart in real life (in km)?" 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:
Line 1: Scale 1 : 50 000 means 1 cm on the map = 50 000 cm in real life.
Line 2: Convert 50 000 cm to metres: 50 000 ÷ 100 = 500 m. Then to km: 500 ÷ 100 = 5 km.
Line 3: Real distance = 4.5 × 5 km = 22.5 km.
Line 4: Final answer: 22.5 km.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: how many metres are in 1 km?)
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? 1 km = 1000 m (not 100 m). So 500 m = 0.5 km, not 5 km.3. Open-ended challenge — design a usable cordial recipe
This question has more than one valid answer. 4 marks
3.1 You're making cordial in the ratio 1 part syrup to 5 parts water for a school stall.
(i) Choose a total volume you want to make, anywhere between 1 L and 5 L. Convert it to mL.
(ii) Work out how much syrup and how much water you need to keep the 1 : 5 ratio.
(iii) Check your two amounts add to your chosen total.
Bonus: Your friend wants the cordial to be HALF as strong (use 1 : 10 instead of 1 : 5) but make the same total volume. How much syrup and water would she use?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Map 1 : 200, 5 cm
1 cm = 200 cm = 2 m. Real distance = 5 × 2 = 10 m.
1.2 — Recipe 4 → 6 people, 200 g sugar
Scale factor = 6/4 = 1.5. Sugar = 200 × 1.5 = 300 g.
1.3 — Drink 1 : 4, total 500 mL
Total parts = 5; 1 part = 100 mL. Juice = 100 mL, water = 400 mL. Check: 100 + 400 = 500 ✓.
1.4 — Photo enlarged 2 : 5
New side = 6 × (5/2) = 6 × 2.5 = 15 cm.
1.5 — House plan 1 : 100
(a) 7 m = 700 cm. Plan length = 700 ÷ 100 = 7 cm.
(b) 3.5 cm × 100 = 350 cm = 3.5 m.
1.6 — Saline 9 : 1, 30 mL salt
Salt (1 part) = 30 mL, so 1 part = 30 mL. Water = 9 × 30 = 270 mL.
2 — Find the mistake
(a) The mistake is on Line 2 (and the wrong "1 cm equals" value is then used in Line 3).
(b) 1 km = 1000 m, not 100 m. So 500 m converted to km is 500 ÷ 1000 = 0.5 km, NOT 500 ÷ 100 = 5 km. The student mixed up the cm-to-m conversion factor with the m-to-km conversion factor.
(c) Corrected working:
Scale 1 : 50 000 means 1 cm on map = 50 000 cm in real life.
Convert: 50 000 cm ÷ 100 = 500 m. Then 500 m ÷ 1000 = 0.5 km.
Real distance = 4.5 × 0.5 km = 2.25 km.
3 — Open-ended challenge (sample solution)
Sample: total = 2.4 L = 2400 mL. Sum of parts = 1 + 5 = 6. 1 part = 2400 ÷ 6 = 400 mL.
Syrup: 1 × 400 = 400 mL. Water: 5 × 400 = 2000 mL. Check: 400 + 2000 = 2400 ✓.
Bonus (1 : 10 instead, same total 2400 mL): Sum of parts = 11. 1 part = 2400 ÷ 11 ≈ 218 mL. Syrup ≈ 218 mL, water ≈ 2182 mL. So you use LESS syrup and MORE water to make the same volume — that's why it tastes weaker.
Other clean totals: 1.2 L (syrup 200 mL, water 1000 mL); 3 L (syrup 500 mL, water 2500 mL); 4.8 L (syrup 800 mL, water 4000 mL).
Marking: 1 mark for a valid total + unit conversion to mL; 1 mark for the correct syrup amount; 1 mark for the correct water amount and sum check; 1 mark for a sensible bonus answer (1 : 10) showing less syrup, more water.