Mathematics • Year 8 • Unit 1 • Lesson 18
Ratio in Context
Apply ratio thinking to map scales, recipe scaling, and mixing solutions. One worked example, one guided example with blanks, then eight independent problems building from clean map distances to multi-step mixes.
1. I do — fully worked example
A map-scale problem worked the long way — every line spells out what the scale means.
Problem. A map has scale 1 : 50 000. Two towns are 4.5 cm apart on the map. How far apart are they really, in kilometres?
Step 1 — Read what the scale means.
1 : 50 000 means 1 cm on the map = 50 000 cm in real life.
Reason: in a scale a : b, "a" measures the map and "b" measures the real world, in the SAME unit.
Step 2 — Convert 50 000 cm to a sensible unit.
50 000 cm = 500 m = 0.5 km
Reason: 100 cm in a metre, 1000 m in a kilometre. So 50 000 cm = 500 m, and 500 m = 0.5 km.
Step 3 — Multiply the map distance by the real-life "1 cm equals" value.
4.5 cm × 0.5 km/cm = 2.25 km
Reason: each map-cm "represents" 0.5 km, so 4.5 map-cm represents 4.5 × 0.5 km.
Step 4 — Sense check.
Real distance should be MUCH bigger than 4.5 cm — and 2.25 km is, so this is sensible.
Reason: when the scale is 1 : (big number), real life is much bigger than the map.
Answer: 2.25 km apart in real life.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A floor plan has scale 1 : 100. A wall is shown as 4.5 cm long on the plan. What is its real length in metres?
Step 1 — Scale 1 : 100 means:
1 cm on plan = ______ cm in real life.
Step 2 — Convert to metres:
______ cm = ______ m. (Use 100 cm = 1 m.)
Step 3 — Multiply plan distance by the "1 cm equals" value:
4.5 cm × ______ m/cm = ______ m
Step 4 — Sense check:
A real wall in a house is a few metres long. Is ______ m sensible? ______
3. You do — independent practice
Show your working. The first four are foundation (single-step map or recipe). The middle two are standard (mixing solutions / scaling recipes). The last two are extension (multi-step or reverse problems).
Foundation — single-step contexts
3.1 Map scale 1 : 100 000. 3 cm on the map = ? km in real life. 1 mark
3.2 A plan has scale 1 : 50. A room is 8 cm long on the plan. What's the real length in metres? 1 mark
3.3 A recipe for 4 people uses 200 g of sugar. How much sugar is needed for 8 people (same ratio)? 1 mark
3.4 A photo is reduced in size in the ratio 5 : 2. A 15 cm side becomes how long? 1 mark
Standard — recipe scaling and drink mixes
3.5 A recipe for 4 people uses 320 g of flour. How much flour is needed for 7 people, keeping the same ratio? (Hint: scale factor = 7/4 = 1.75.) 2 marks
3.6 A cordial is mixed in the ratio 1 part syrup to 7 parts water. How much syrup is needed to make 480 mL of cordial total? 2 marks
Extension — multi-step / reverse problems
3.7 A drink is mixed in the ratio 1 : 4 (juice : water). You want to make exactly 500 mL of drink. How much juice and how much water do you need? 2 marks
3.8 A floor plan uses scale 1 : 50. A real wall is 6 m long. (a) Convert 6 m to cm. (b) What length should it be drawn as on the plan? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded plan 1 : 100, wall 4.5 cm)
Step 1: 1 cm on plan = 100 cm in real life.
Step 2: 100 cm = 1 m.
Step 3: 4.5 cm × 1 m/cm = 4.5 m.
Step 4: 4.5 m is a sensible length for a real wall. Yes, sensible.
3.1 — Map 1 : 100 000, 3 cm
1 cm = 100 000 cm = 1000 m = 1 km. So 3 cm = 3 km.
3.2 — Plan 1 : 50, 8 cm
1 cm = 50 cm = 0.5 m. So 8 cm × 0.5 m/cm = 4 m. (Or: 8 × 50 = 400 cm = 4 m.)
3.3 — Recipe scaled 4 → 8 people
Scale factor = 8 ÷ 4 = 2. Sugar = 200 × 2 = 400 g.
3.4 — Photo reduced 5 : 2
New side = 15 × (2/5) = 15 × 0.4 = 6 cm.
3.5 — Recipe 4 → 7 people, 320 g flour
Scale factor = 7/4 = 1.75. Flour = 320 × 1.75 = 560 g. Sanity check: 560 ÷ 7 = 80 g/person, same as 320 ÷ 4 = 80 g/person ✓.
3.6 — Cordial 1 : 7, total 480 mL
Total parts = 1 + 7 = 8. 1 part = 480 ÷ 8 = 60 mL. Syrup (1 part) = 60 mL. (Water = 420 mL; check: 60 + 420 = 480 ✓.)
3.7 — Juice/water 1 : 4, total 500 mL
Total parts = 5. 1 part = 500 ÷ 5 = 100 mL. Juice = 100 mL, water = 400 mL. Check: 100 + 400 = 500 ✓.
3.8 — Plan 1 : 50, real 6 m
(a) 6 m = 600 cm.
(b) Plan length = 600 ÷ 50 = 12 cm.