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Units, scales and compound measures

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Converting Metric Units

  • Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. To convert, multiply or divide by the correct power of 10.
  • Mass: 1 tonne = 1000 kg, 1 kg = 1000 g, 1 g = 1000 mg. Use the same decimal shift method as for length.
  • Capacity: 1 litre = 1000 ml, and 1 ml = 1 cm3. Also 1 m3 = 1000 litres.
  • Time: 1 hour = 60 minutes, 1 minute = 60 seconds. Converting time often involves multiplying or dividing by 60, not by 10.
  • For an illustrative exchange rate £1 = $1.25, multiply a pound amount by 1.25 to obtain dollars. Use the rate given by the question.
  • Area: When converting squared units, square the linear conversion factor. For example, 1 m2 = (100 cm)2 = 10000 cm2.
  • Volume: When converting cubed units, cube the linear conversion factor. For example, 1 m3 = (100 cm)3 = 1,000,000 cm3.

Metres to centimetres: factor 100 per dimension

Metres to centimetres: factor 100 per dimensionQuantityLinear factor powerEquivalent1 m100100 cm1 m²100²10000 cm²1 m³100³1000000 cm³

Compound Measures: Speed

  • Speed is a compound measure: speed = distance ÷ time, written as S = \frac{D}{T} (S = speed, D = distance, T = time).
  • Common units: m/s, km/h, mph. Always ensure distance and time units match the speed unit required.
  • To convert m/s to km/h, multiply by 3.6 (because 1 m/s = 3600 m per hour = 3.6 km/h).
  • To convert km/h to m/s, divide by 3.6.
  • Rearrange the formula: distance = speed × time, and time = distance ÷ speed.
  • When solving, convert all values to the same unit system first, e.g. change km to m before using m/s.

Compound Measures: Density

  • Density = mass ÷ volume, written as D = \frac{M}{V} (D = density, M = mass, V = volume).
  • Common units are g/cm³ and kg/m³. Liquid water has density about 1 g/cm³ under ordinary conditions; density varies with material and conditions.
  • Rearrange: mass = density × volume, and volume = mass ÷ density.
  • When mixing units (e.g. mass in kg and volume in cm3), convert so that both are consistent before calculating.
  • A common error is inverting the formula: remember density is mass per unit volume, so larger mass in same volume means higher density.

Compound Measures: Pressure

  • Pressure equals normal force divided by area: P = F/A.
  • Common units: N/m2 (also called pascal, Pa), N/cm2.
  • Rearrange: force = pressure × area, and area = force ÷ pressure.
  • Ensure force and area units are compatible: if area is in cm2, force in N gives pressure in N/cm2.
  • Pressure is often used in contexts like tyres, hydraulics, and atmospheric pressure.

Converting Compound Units

  • To convert a compound unit like m/s to km/h, convert the numerator and denominator separately: 1 m = 0.001 km, 1 s = 1/3600 h, so 1 m/s = 0.001 km ÷ (1/3600 h) = 3.6 km/h.
  • For density, converting g/cm3 to kg/m3: multiply by 1000 (since 1 g = 0.001 kg and 1 cm3 = 10-6 m3, so 1 g/cm3 = 0.001 kg / 10-6 m3 = 1000 kg/m3).
  • Always write down the conversion factors for each unit and combine them carefully.
  • Avoid rounding intermediate steps; keep full precision until the final answer.
  • Check that the final units make sense: for example, a speed of 100 km/h is about 27.8 m/s.

Scale Diagrams and Maps

  • A scale on a map or diagram shows the relationship between a length on the drawing and the real length.
  • Ratio scale like 1:25000 means 1 unit on the map represents 25000 of the same units in real life.
  • To find real distance from map distance, multiply map distance by the scale factor.
  • To find map distance from real distance, divide real distance by the scale factor.
  • Convert units if needed: for a 1:25000 map, 1 cm on the map = 25000 cm = 250 m in reality.
  • Scale factors can be written as ratios, as a fraction (e.g. 1/25000), or as a statement like '1 cm to 250 m'.

Map scale 1:50,000

Map scale 1:50,000Map / cmReal / mReal / km15000.5210001840004

Using Scale Factors

  • A scale factor is the multiplier that converts a length on a scale drawing to the real length.
  • For example, a scale of 1:100 means every length on the drawing is 1/100 of the real length, so real length = drawing length × 100.
  • Scale factors also apply to areas and volumes: area scale factor = (linear scale factor)2, volume scale factor = (linear scale factor)3.
  • If a map has scale 1:50000, an area of 2 cm2 on the map represents 2 × (50000)2 cm2 in reality.
  • When solving problems, always state the units clearly and convert to a consistent unit before calculating.

Common Misconceptions and Errors

  • Area conversion error: 1 m2 is 10000 cm2, not 100 cm2. Remember to square the linear factor.
  • Volume conversion error: 1 m3 is 1,000,000 cm3, not 1000 cm3. Cube the linear factor.
  • Mixing units: In speed problems, don't mix km with m/s without converting. Always convert to the same unit system.
  • Inverting formulas: For density, mass = density × volume, not mass = density ÷ volume. Check the units: density × volume gives mass.
  • Misreading scale: A scale of 1:25000 means 1 cm = 250 m, not 25 m. Watch the number of zeros.
  • Rounding too early: In multi-step conversions, keep full precision until the final answer to avoid errors.

Problem-Solving Strategies

  • Identify the compound measure involved (speed, density, pressure) and write down the formula.
  • List the given quantities and the required quantity, then decide whether to convert units first.
  • Convert all units to the same system before substituting into the formula.
  • For map scale problems, write the scale as a fraction or ratio and set up a proportion.
  • Check reasonableness against the context. For example, 50 m/s equals 180 km/h: high for an ordinary road journey, but not automatically a conversion error.
  • Always include units in your final answer.

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  1. 1.Which of the following is the correct conversion factor between metres and kilometres?

    Easy
    • A1 km = 100 m
    • B1 km = 1000 m
    • C1 km = 10 000 m
    • D1 km = 100 000 m

  2. 2.How many square centimetres are there in 1 square metre?

    Easy
    • A100
    • B1000
    • C10 000
    • D100 000

  3. 3.The density of a substance is its mass divided by its volume.

    Easy

    True or false?

  4. 4.A block has mass 200 g and volume 50 cm3. What is its density in g/cm3?

    Medium
    • A0.25
    • B4
    • C250
    • D10 000

  5. 5.A force of 20 N acts on an area of 4 m2. What is the pressure?

    Medium
    • A5 Pa
    • B80 Pa
    • C0.2 Pa
    • D16 Pa

  6. 6.Which of the following are equivalent to 1 kg? (Select all that apply)

    Medium
    • A1000 g
    • B100 g
    • C0.001 tonnes
    • D1 000 000 mg

  7. 7.Which of the following is the correct relationship between pressure, force and area?

    Medium
    • AP = F × A
    • BP = F / A
    • CP = A / F
    • DP = F + A

  8. 8.Match each compound measure to its standard unit.

    Medium
    • Speed
    • Density
    • Pressure
    • m/s
    • kg/m3
    • Pa

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