To compare prices fairly, find the cost per unit (per gram, ml, litre, item).
📌 Formula: Unit Price = Total Price ÷ Quantity
Example: 600 g of pasta costs £1.80 Unit price = £1.80 ÷ 600 = £0.003 per gram = 0.3p per gram
Example: 1.5 litres of juice costs £1.20 Unit price = £1.20 ÷ 1500 ml = £0.0008 per ml = 0.08p per ml
💡 Always use the same unit when comparing. Convert kg → g, litres → ml first.
Part 2: Best Value Comparisons
Compare unit prices: the lower unit price = better value.
Pack A: 400 g for £2.00 → £2.00 ÷ 400 = 0.5p/g
Pack B: 750 g for £3.30 → £3.30 ÷ 750 = 0.44p/g
Pack B is better value (lower price per gram) ✅
⚠️ Watch out! A bigger pack is NOT always better value. Always check the unit price.
💡 "Buy 3 for £5" — find unit price: £5 ÷ 3 = £1.67 per item. Compare with single price.
Part 3: Rate Calculations
A rate compares two different quantities, e.g. km per hour, litres per minute.
📌 Speed: Speed = Distance ÷ Time · Distance = Speed × Time · Time = Distance ÷ Speed
Example: A car travels 240 km in 3 hours → Speed = 240 ÷ 3 = 80 km/h
📌 Flow rate: Rate = Volume ÷ Time A tank fills 300 litres in 5 minutes → Flow rate = 300 ÷ 5 = 60 litres/min
📌 Pay rate: Hourly pay = Total pay ÷ Hours worked £112 for 8 hours → £112 ÷ 8 = £14/hour
💡 Rate = Amount ÷ Time — it always tells you how much per one unit of time (or distance).
Part 4: Converting Rates
To convert km/h to m/s: multiply by 1000 (km→m) then divide by 3600 (h→s).
📌 km/h → m/s: divide by 3.6 (or × 1000 ÷ 3600)
📌 m/s → km/h: multiply by 3.6
Example: 90 km/h → m/s: 90 ÷ 3.6 = 25 m/s
Example: 15 m/s → km/h: 15 × 3.6 = 54 km/h
💡 For other rates: convert numerator unit, then denominator unit, separately.
Example: 5 litres/min → ml/s: 5000 ml ÷ 60 s = 83.3 ml/s
Part 5: Scaling Rates (Worker Problems)
When more workers share a job, it takes less time — this is inverse proportion.
📌 Method: find total work units (workers × time), then divide by new number of workers.
Example: 3 workers take 8 hours. How long for 6 workers? Total work = 3 × 8 = 24 worker-hours Time for 6 workers = 24 ÷ 6 = 4 hours
Example: A machine fills 2000 bottles in 5 hours. Rate = 400 bottles/h. Two machines → 800 bottles/h → 2000 ÷ 800 = 2.5 hours
⚠️ Key rule: Workers × Time = constant (as long as each worker works at the same rate).
💡 Worked Examples
Example 1: Finding Unit Price
Cereal: 750 g for £2.85 or 1.2 kg for £4.20. Which is better value?
Step 1: Convert to same unit — 1.2 kg = 1200 g
Step 2: Unit price A = £2.85 ÷ 750 = £0.0038/g = 0.38p per gram
Step 3: Unit price B = £4.20 ÷ 1200 = £0.0035/g = 0.35p per gram
Answer: The 1.2 kg box (Pack B) is better value ✅
💡 0.35 < 0.38, so fewer pence per gram = better value.
Example 2: Speed Calculation
A cyclist travels 48 km at a constant speed of 16 km/h. How long does the journey take?
Step 1: Use Time = Distance ÷ Speed
Step 2: Time = 48 ÷ 16 = 3 hours
Check: 16 km/h × 3 h = 48 km ✅
🚴 Use the DST triangle: cover what you want — D = S × T, S = D ÷ T, T = D ÷ S
Example 3: Converting km/h to m/s
A train travels at 108 km/h. Convert to m/s.
Step 1: 108 km/h = 108 × 1000 m per 3600 s
Step 2: = 108 000 ÷ 3600 = 30 m/s
Shortcut: 108 ÷ 3.6 = 30 m/s ✅
💡 Dividing by 3.6 always converts km/h → m/s.
Example 4: Scaling Workers
5 painters take 12 days to paint a building. The deadline is in 6 days. How many painters are needed?
Step 1: Total work = 5 × 12 = 60 painter-days
Step 2: Painters needed = Total work ÷ Available days = 60 ÷ 6 = 10 painters
Check: 10 painters × 6 days = 60 painter-days ✅
⚠️ More workers = less time. This is inverse proportion.
🛒 Interactive Visualizer
Supermarket Shelf — Click the Best Value! 🏆
Calculate the unit price (p/g or p/ml) for each product, then click the best value one.
Rate Race — Who Finishes First? 🏁
Each worker/machine has a different rate. A job needs 600 items to be completed. Who finishes first?
Rate Converter — km/h ↔ m/s 🔄
→
Spot the Deal — Click the Better Value! 🎯
In each pair, click the item with the lower unit price.
✏️ Exercise 1: Unit Price
Find the unit price. Give answers to 2 decimal places (in pence) where needed.
1. 200 g of cheese costs £1.60. Price per gram =
p/g
2. 500 ml of shampoo costs £3.50. Price per ml =
p/ml
3. A pack of 8 pens costs £2.40. Price per pen =
p
4. 1.5 kg of rice costs £1.80. Price per 100 g =
p
5. 2 litres of orange juice costs £1.30. Price per 100 ml =
p
6. 6 eggs cost £1.74. Price per egg =
p
7. 750 ml of cooking oil costs £2.25. Price per ml =
p/ml
8. A 12-pack of water bottles costs £4.20. Price per bottle =
p
✏️ Exercise 2: Best Value Comparisons
Type A or B for the better value pack.
1. Yoghurt: A 500 g for £1.40 | B 750 g for £1.95
Better value:
2. Butter: A 250 g for £1.60 | B 400 g for £2.40
Better value:
3. Shampoo: A 300 ml for £2.10 | B 500 ml for £3.20
Better value:
4. Crisps: A 6-pack for £2.10 | B 5-pack for £1.60
Better value:
5. Tomatoes: A 400 g for £1.20 | B 1 kg for £2.70
Better value:
6. Mineral water: A 2 litres for £0.85 | B 6 × 500 ml for £2.40
Better value:
7. Fuel: petrol at A £1.56/litre vs diesel at B £1.62/litre (car does 45 mpg petrol or 55 mpg diesel). Per mile cheaper:
Cheaper per mile:
A: 1.40÷500 = 0.28p/g; B: 1.95÷750 = 0.26p/g → B
A: 1.60÷250 = 0.64p/g; B: 2.40÷400 = 0.60p/g → B
A: 2.10÷300 = 0.70p/ml; B: 3.20÷500 = 0.64p/ml → B
A: 2.10÷6 = 35p/pack; B: 1.60÷5 = 32p/pack → B
A: 1.20÷400 = 0.30p/g; B: 2.70÷1000 = 0.27p/g → B
A: 85p÷2000ml = 0.0425p/ml; B: 240p÷3000ml = 0.08p/ml → A
1 mile petrol: £1.56÷45 = 3.47p; diesel: £1.62÷55 = 2.95p → B (diesel cheaper per mile)
✏️ Exercise 3: Rate Calculations
1. A car travels 180 km in 2.5 hours. Speed =
km/h
2. A runner completes 10 km in 50 minutes. Speed =
km/h
3. A tap fills 240 litres in 8 minutes. Flow rate =
litres/min
4. A worker earns £126 for 9 hours. Hourly rate = £
5. A train travels at 90 km/h for 2 hours 30 minutes. Distance =
km
6. A pump empties 600 litres at 40 litres/min. Time =
minutes
7. A factory produces 1440 items in 6 hours. Production rate =
items/hour
8. A cyclist travels 54 km at 18 km/h. Time =
hours
🔄 Exercise 4: Converting Rates
1. Convert 36 km/h to m/s. Answer =
m/s
2. Convert 25 m/s to km/h. Answer =
km/h
3. Convert 54 km/h to m/s. Answer =
m/s
4. Convert 20 m/s to km/h. Answer =
km/h
5. Convert 3 litres/min to ml/s. Answer =
ml/s
6. Convert £15/hour to pence/minute. Answer =
p/min
7. Convert 120 km/h to m/s. Answer =
m/s (1 d.p.)
8. A car does 40 miles per gallon. 1 gallon = 4.546 litres. Convert to km/l (1 mile = 1.609 km). Answer =
km/l (1 d.p.)
36 ÷ 3.6 = 10 m/s
25 × 3.6 = 90 km/h
54 ÷ 3.6 = 15 m/s
20 × 3.6 = 72 km/h
3000 ml ÷ 60 s = 50 ml/s
1500p ÷ 60 = 25 p/min
120 ÷ 3.6 = 33.3 m/s
40 × 1.609 ÷ 4.546 = 64.36 ÷ 4.546 = 14.2 km/l
📝 Exercise 5: Mixed Word Problems
1. 4 workers take 15 days to build a wall. How long will 6 workers take?
Answer: days
2. A supermarket sells orange juice: 1 litre for £1.20 or 2.5 litres for £2.75. Which is better value and by how much per litre?
Better value: Saving per litre: p
3. A car travels at 108 km/h. Express this speed in m/s.
Answer: m/s
4. Ali earns £13.50 per hour and works 37.5 hours per week. Beth earns £520 per week for 40 hours. Who has a higher hourly rate?
Higher rate:
5. 8 machines produce 2400 units per day. How many machines are needed to produce 4500 units per day?
Answer: machines
6. A pool holds 36 000 litres. Pump A fills it at 120 litres/min. Pump B fills it at 80 litres/min. If both run together, how long to fill the pool? Give your answer in hours.
Answer: hours
7. Petrol costs £1.62 per litre. A car's fuel efficiency is 12 km per litre. How much does it cost to drive 300 km?
Answer: £
8. 3 workers take 6 hours. If one worker leaves after 2 hours, how long in total does the job take from the start?
Answer: hours
Total = 4 × 15 = 60 worker-days; Time = 60 ÷ 6 = 10 days
1 litre: 120p; 2.5 litres: 275p → 110p/litre. 2.5 litre is better by 10p/litre