Less common Papers 1 & 2

Units: Compound Units (Speed, Density, Pressure) and Converting Between Them

A compound unit like m/s or kg/m3 is a formula in disguise: the unit itself tells you how the quantity is built from length, mass and time. Converting between related units means rewriting each base unit in the target form and cancelling the numbers exactly, usually down to a tidy whole number that fits one of the options.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • A compound unit is read literally as a formula: m/s means metres per second, i.e. distance/time, and kg/m3 means mass/volume.
  • The three standard rates are speed = distance/time (m/s), density = mass/volume (kg/m3 or g/cm3), and pressure = force/area.
    N/m2 = Pa
  • To convert a compound unit, change the numerator unit and the denominator unit separately, then simplify the resulting number.
  • Speed converts by a fixed factor: 1 m/s = 3.6 km/h, so multiply by 3.6 going m/s → km/h and divide by 3.6 going km/h → m/s.
  • Area conversions square the length factor and volume conversions cube it.
    1 m2 = 104 cm2, 1 m3 = 106 cm3
  • Density in g/cm3 becomes kg/m3 by multiplying by 1000; e.g. water is 1 g/cm3 = 1000 kg/m3.
  • Keep exact fractions (like 1/3600) instead of decimals - the arithmetic on this paper almost always cancels neatly.
  • Sanity-check by unit size: a smaller unit produces a larger number, so a km/h value is about 3.6 times its m/s value.

Formulae

speed = distance / time

Rate-of-travel problems; the unit m/s or km/h signals distance over time.

density = mass / volume

When two of mass, volume and density are known; units g/cm3 or kg/m3.

pressure = force / area

Force spread over a surface; units N/m2, also called pascals (Pa).

1 m/s = 3.6 km/h

Switching between the two common speed units - multiply by 3.6 one way, divide by 3.6 the other.

1 g/cm3 = 1000 kg/m3

Converting a density between the g/cm3 and kg/m3 systems.

1 m2 = 104 cm2, 1 m3 = 106 cm3

Area and volume conversions - square or cube the length factor rather than using it once.

Definitions

Base unit
A fundamental measurement unit - metre for length, second for time, gram or kilogram for mass - from which other units are constructed.
Compound unit
A unit built by combining base units through multiplication or division, such as m/s, kg/m3 or N/m2.
Density
Mass per unit volume of a substance, given by density = mass/volume; common units are g/cm3 and kg/m3.
Conversion factor
The number you multiply by to change from one related unit to another, e.g. 1000 for km → m or 3600 for hours → seconds.

Worked examples

1

Without a calculator, convert a speed of 54 km/h into metres per second.

  1. 1

    Rewrite each unit in base form:

    54 km/h = (54 × 1000 m)/(3600 s)
  2. 2

    Group the numbers:

    value = 54000/3600 m/s
  3. 3

    Cancel zeros then divide:

    value = 540/36 = 15 m/s

Answer: 15 m/s

2

A force of 200 N is applied to a surface of area 25 cm2. Find the pressure in N/m2 (pascals).

  1. 1

    Each cm is 1/100 m, so square it:

    1 cm2 = 10-4 m2
  2. 2

    Convert the given area:

    A = 25 × 10-4 = 0.0025 m2
  3. 3

    Pressure is force divided by area:

    P = 200/0.0025
  4. 4

    Divide (200/0.0025 = 200 × 400):

    P = 80000 N/m2

Answer: 80000 Pa (= 80000 N/m2)

Common mistakes

  • ×Forgetting to square or cube the length factor, e.g. using 100 instead of 1002 = 104 when going from cm2 to m2 (or 106 for volumes).
  • ×Converting only the numerator of a compound unit and leaving the denominator unchanged, such as changing km to m but forgetting to change hours to seconds.
  • ×Multiplying when you should divide: moving to a larger unit makes the number smaller, so km/h to m/s is divide by 3.6, not multiply.
  • ×Mixing inconsistent units before computing, e.g. combining a mass in grams with a volume in m3 without converting first.
  • ×Rounding partway through on a no-calculator question instead of cancelling exact fractions to a whole number.

No-calculator tips

  • ✓Rewrite every unit in the target units first, then do a single clean division - the numbers usually cancel to something tidy.
  • ✓Memorise the anchor factors: 3.6 (m/s → km/h), 1000 (g/cm3 → kg/m3), 104 (m2 → cm2), 106 (m3 → cm3).
  • ✓Cancel zeros before dividing: 54000/3600 → 540/36 → 15.
  • ✓Write the units as fractions and cancel them like algebra to confirm your conversion factor is the right way up.
  • ✓Estimate to eliminate options: a metal's density in kg/m3 runs into the thousands, so any choice under about 100 must be wrong.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.The density of a metal is 8 g/cm3. Expressed in kg/m3, this density is:

  • A. 0.008
  • B. 0.8
  • C. 800
  • D. 8000
  • E. 8000000
Show answer

Answer: D — 8000

1 g = 10-3 kg and 1 cm3 = 10-6 m3, so 1 g/cm3 = 103 kg/m3; hence 8 g/cm3 = 8000 kg/m3. The 8000000 trap uses the 106 volume factor but forgets to convert grams to kilograms.

Q2.A cyclist travels at a steady 15 m/s. What is this speed in km/h?

  • A. 4.2
  • B. 54
  • C. 540
  • D. 54000
Show answer

Answer: B — 54

To go from m/s to km/h multiply by 3.6, so 15 × 3.6 = 54 km/h. Dividing by 3.6 (giving 4.2) applies the conversion in the wrong direction.

Q3.A pressure is measured as 2 N/cm2. Written in pascals (Pa = N/m2), this pressure is:

  • A. 0.0002
  • B. 200
  • C. 20000
  • D. 2000000
Show answer

Answer: C — 20000

1 m2 = 1002 = 104 cm2, so N/cm2 → N/m2 means multiplying by 104: 2 × 104 = 20000 Pa. The 2000000 trap wrongly uses the 106 factor for cm3 → m3 instead of the area factor 104.

Read this topic in the official UAT-UK TMUA content specification →

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