N13: Units & Conversions
Use standard units of measure; compound measures; metric/imperial conversions
📋 Key Concepts
Metric System: Based on powers of 10. The standard units are metre (length), gram (mass), and litre (capacity).
Metric Prefixes
| Prefix | Symbol | Meaning |
|---|---|---|
| kilo- | k | × 1000 |
| centi- | c | ÷ 100 |
| milli- | m | ÷ 1000 |
📝 Length Conversions
Length Units:
1 km = 1000 m
1 m = 100 cm
1 m = 1000 mm
1 cm = 10 mm
1 km = 1000 m
1 m = 100 cm
1 m = 1000 mm
1 cm = 10 mm
Example 1
Convert 3.5 km to metres
Solution: 3.5 × 1000 = 3500 m
Example 2
Convert 4500 mm to metres
Solution: 4500 ÷ 1000 = 4.5 m
📝 Mass Conversions
Mass Units:
1 tonne = 1000 kg
1 kg = 1000 g
1 g = 1000 mg
1 tonne = 1000 kg
1 kg = 1000 g
1 g = 1000 mg
Example 3
Convert 2.4 kg to grams
Solution: 2.4 × 1000 = 2400 g
📝 Volume/Capacity Conversions
Volume Units:
1 litre = 1000 ml
1 ml = 1 cm³
1 litre = 1000 cm³
1 m³ = 1000 litres
1 litre = 1000 ml
1 ml = 1 cm³
1 litre = 1000 cm³
1 m³ = 1000 litres
Example 4
Convert 2.8 litres to millilitres
Solution: 2.8 × 1000 = 2800 ml
📝 Time Conversions
Time Units:
1 minute = 60 seconds
1 hour = 60 minutes
1 day = 24 hours
1 week = 7 days
1 year = 365 days (366 in leap year)
1 minute = 60 seconds
1 hour = 60 minutes
1 day = 24 hours
1 week = 7 days
1 year = 365 days (366 in leap year)
Example 5
Convert 2.5 hours to minutes
Solution: 2.5 × 60 = 150 minutes
Example 6
Convert 200 minutes to hours and minutes
Solution: 200 ÷ 60 = 3 remainder 20
Answer: 3 hours 20 minutes
📝 Metric-Imperial Conversions
You should know these approximate conversions:
1 inch ≈ 2.5 cm
1 foot ≈ 30 cm
1 mile ≈ 1.6 km (or 8 km ≈ 5 miles)
1 pound (lb) ≈ 450 g
1 kg ≈ 2.2 lb
1 gallon ≈ 4.5 litres
1 pint ≈ 570 ml
1 inch ≈ 2.5 cm
1 foot ≈ 30 cm
1 mile ≈ 1.6 km (or 8 km ≈ 5 miles)
1 pound (lb) ≈ 450 g
1 kg ≈ 2.2 lb
1 gallon ≈ 4.5 litres
1 pint ≈ 570 ml
Example 7
Convert 5 miles to kilometres
Solution: 5 × 1.6 = 8 km
Or: 5 miles = 8 km (remember 8 km ≈ 5 miles)
Example 8
Convert 12 inches to cm
Solution: 12 × 2.5 = 30 cm
Note: 12 inches = 1 foot ≈ 30 cm
📝 Compound Units
Compound units: Combine two or more measurements, such as speed, density, and pressure.
| Quantity | Formula | Units |
|---|---|---|
| Speed | Speed = Distance ÷ Time | km/h, m/s, mph |
| Density | Density = Mass ÷ Volume | kg/m³, g/cm³ |
| Pressure | Pressure = Force ÷ Area | N/m² (Pascals) |
| Rate of pay | Pay = Rate × Time | £/hour |
Example 9
A car travels 180 km in 2.5 hours. Find its average speed.
Solution:
Speed = Distance ÷ Time
Speed = 180 ÷ 2.5 = 72 km/h
Example 10
A block has mass 240 g and volume 80 cm³. Find its density.
Solution:
Density = Mass ÷ Volume
Density = 240 ÷ 80 = 3 g/cm³
🧮 Worked Examples — Exam Style
Metric conversions are powers of ten. Imperial conversions (miles, pounds, gallons, stones) are GIVEN in the question — the skill is choosing multiply or divide.
Convert 55 miles to km (1 mile = 1.6 km).
Km per mile, so MULTIPLY: 55 × 1.6 = 88 km
Reasoning check: a mile is LONGER than a km, so the km number must be bigger.
M1 for × 1.6, A1 for 88. The 'bigger unit → bigger number' check catches most direction errors.
Convert 250 g to ounces (1 ounce = 28 g). Give your answer to 1 decimal place.
Grams per ounce given, so DIVIDE: 250 ÷ 28 = 8.928…
≈ 8.9 ounces
M1 for ÷ 28, A1 for 8.9. When the conversion says '1 ounce = 28 g', ounces → divide.
Exam wording to expect: the conversion factor is always given for imperial units — look for "use 1 mile = …".
❓ Practice Questions
Q1: Convert 4500 grams to kg
Q2: Convert 3.2 km to metres
Q3: Convert 90 minutes to hours
Q4: Convert 10 miles to km (use 1 mile = 1.6 km)
Q5: A cyclist travels 48 km in 2 hours. Find the average speed.
Q6: Convert 2.5 gallons to litres (1 gallon = 4.5 litres)
Q7: Convert 36.6°C… no — convert 5 stones to kg (1 stone = 6.35 kg)
✅ Answers
- 4500 ÷ 1000 = 4.5 kg
- 3.2 × 1000 = 3200 m
- 90 ÷ 60 = 1.5 hours
- 10 × 1.6 = 16 km
- 48 ÷ 2 = 24 km/h
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. g → kg: ÷ 1000 → 4.5 kg
Q2. km → m: × 1000 → 3200 m
Q3. 90 ÷ 60 = 1.5 hours
Q4. 10 × 1.6 = 16 km
Q5. Speed = 48 ÷ 2 = 24 km/h
Q6. 2.5 × 4.5 = 11.25 litres
Q7. 5 × 6.35 = 31.75 kg
📌 Key Facts to Memorise
- Metric: km-m (×1000), m-cm (×100), kg-g (×1000), litre-mL (×1000)
- Imperial facts given in questions: 1 mile ≈ 1.6 km · 1 kg ≈ 2.2 lb · 1 inch ≈ 2.5 cm · 1 gallon ≈ 4.5 litres · 1 stone = 14 lb
- Bigger unit → SMALLER number when converting up (88 km is 55 miles)
- Area: square the factor; Volume: cube it
- Speed units: km/h ÷ 3.6 = m/s
🗺️ In the Exam
Conversion questions: 1–2 marks, and embedded in speed/density/recipe questions. The given-factor imperial conversions are Foundation staples on Papers 2/3.
Links to: units and measure (G14), compound measures (R1), scale drawings (R2).
⚡ Targeted Drills
Cover the answers, work each one, then mark yourself. If any takes over a minute, revisit the matching section above.
- 3.4 kg → g → 3400 g · 720 mL → litres → 0.72 L
- 45 min → hours (decimal) → 0.75 h · 2.2 h → hours+min → 2 h 12 min
- 12 miles → km (× 1.6) → 19.2 km · 40 km → miles (÷ 1.6) → 25 miles
- 6 ounces → g (× 28) → 168 g
- Speed 63 km/h in m/s (÷ 3.6) → 17.5 m/s
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- For metric conversions: big unit → small unit = multiply; small → big = divide
- Memorise the key conversions (especially metric-imperial)
- Time calculations often need mixed units (hours and minutes)
- Check units carefully - cm³ is not the same as cm!
- In compound unit questions, always state the units in your answer
🧠 Problem-Solving Strategies
Problem-Solving
For multi-step unit conversions, convert one step at a time and write down each step. For compound measures, always check units are compatible before calculating (e.g., don't mix km and m). When converting between metric and imperial, learn the key approximations (5 miles ≈ 8 km, 1 kg ≈ 2.2 lb).
Multi-Step Problem
A car travels at 55 mph. Convert this speed to km/h. (Use 5 miles ≈ 8 km)
Solution: 55 miles = 55 × 8/5 = 88 km. So 55 mph ≈ 88 km/h.
⚠️ Common Errors
Watch Out!
1. Wrong: 2.5 hours = 2 hours 50 minutes (treating 0.5 as 50) Correct: 0.5 hours = 30 minutes, so 2.5 hours = 2 hours 30 minutes
2. Wrong: Converting cm² to m² by dividing by 100 Correct: Area units divide by 100² = 10,000. So 500 cm² = 0.05 m²
3. Wrong: Speed = Distance × Time (multiplying instead of dividing) Correct: Speed = Distance ÷ Time (or use the triangle: Distance on top, Speed and Time on the bottom)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A runner completes a 10 km race in 42 minutes 30 seconds. (a) Convert the distance to metres and the time to seconds. (b) Calculate the average speed in m/s. (c) Convert this speed to km/h. Give your answer to 1 decimal place.
(a) 10 km = 10,000 m. Time = (42 × 60) + 30 = 2520 + 30 = 2550 seconds.
(b) Speed = 10,000 ÷ 2550 = 3.92 m/s (to 2 d.p.).
(c) 3.92 m/s = 3.92 × 3600 ÷ 1000 = 14.12 km/h ≈ 14.1 km/h.
Mark scheme: 1 mark for distance conversion, 1 mark for time conversion, 2 marks for speed in m/s, 2 marks for conversion to km/h
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A recipe from the US uses 2 cups of flour and 3 pints of milk. A UK cook has 500 g of flour and 2 litres of milk.
(a) 1 US cup of flour ≈ 120 g. Does the cook have enough flour?
(b) 1 US pint ≈ 470 ml. Does the cook have enough milk?
(c) The cook says "I can just double the recipe to use all my ingredients." Is this a sensible strategy? Explain.
Answers: (a) 2 cups = 2 × 120 = 240 g needed. Has 500 g — yes, enough. (b) 3 US pints = 3 × 470 = 1410 ml = 1.41 litres. Has 2 litres — yes, enough. (c) Doubling would need 480 g flour and 2.82 litres milk. The cook has 500 g flour (enough) but only 2 litres milk (not enough for 2.82 L). So no, this would not work — there isn't enough milk.
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🎬 Video Resources
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