9.1 Units, Scales & Approximate Imperial Equivalents

Measurement questions form a cornerstone of the ISEB Common Pre-Test Mathematics syllabus. In 11+ entrance examinations, measurement problems rarely test simple recall in isolation; instead, they embed dimensional calculations within complex, multi-step word problems requiring unit conversions, proportion, financial reasoning, and spatial interpretation. Success demands absolute fluency with both decimal metric conversions and foundational imperial benchmarks, alongside systematic methods for calculating elapsed time, interpreting transit timetables, and reading unlabelled scale divisions.

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1. The Metric System & Unit Conversions

The metric system is built upon a base-10 decimal framework where prefixes indicate multiplication or division by powers of 10:

• milli- ($\times 0.001$ or $\div 1{,}000$)
• centi- ($\times 0.01$ or $\div 100$)
• deci- ($\times 0.1$ or $\div 10$)
• kilo- ($\times 1{,}000$)

Length Conversions

Length measures one-dimensional distance. The fundamental metric units tested are millimetres ($\text{mm}$), centimetres ($\text{cm}$), metres ($\text{m}$), and kilometres ($\text{km}$):

$$\begin{aligned} 1\text{ cm} &= 10\text{ mm} \\ 1\text{ m} &= 100\text{ cm} = 1{,}000\text{ mm} \\ 1\text{ km} &= 1{,}000\text{ m} = 100{,}000\text{ cm} = 1{,}000{,}000\text{ mm} \end{aligned}$$

To convert from a larger unit to a smaller unit, you multiply by the conversion factor (e.g., $3.4\text{ m} \times 100 = 340\text{ cm}$). To convert from a smaller unit to a larger unit, you divide by the conversion factor (e.g., $450\text{ m} \div 1{,}000 = 0.45\text{ km}$).

┌────────────────────────────────────────────────────────┐
│               Metric Length Conversions                │
│                                                        │
│   km  ──── ( × 1,000 ) ───►  m  ──── ( × 100 ) ───► cm │
│   km  ◄─── ( ÷ 1,000 ) ────  m  ◄─── ( ÷ 100 ) ──── cm │
│                                                        │
│                 cm  ──── ( × 10 ) ───► mm              │
│                 cm  ◄─── ( ÷ 10 ) ──── mm              │
└────────────────────────────────────────────────────────┘

Mass Conversions

Mass measures the amount of matter in an object. In Key Stage 2 mathematics, the three core units are grams ($\text{g}$), kilograms ($\text{kg}$), and tonnes ($\text{t}$):

$$\begin{aligned} 1\text{ kg} &= 1{,}000\text{ g} \\ 1\text{ tonne} &= 1{,}000\text{ kg} = 1{,}000{,}000\text{ g} \end{aligned}$$

Candidates must be comfortable working with fractional parts of a kilogram:

• $\frac{1}{2}\text{ kg} = 500\text{ g} = 0.5\text{ kg}$
• $\frac{1}{4}\text{ kg} = 250\text{ g} = 0.25\text{ kg}$
• $\frac{3}{4}\text{ kg} = 750\text{ g} = 0.75\text{ kg}$
• $\frac{1}{8}\text{ kg} = 125\text{ g} = 0.125\text{ kg}$
• $\frac{1}{10}\text{ kg} = 100\text{ g} = 0.1\text{ kg}$

Capacity & Liquid Volume Conversions

Capacity represents the internal volume of a vessel that can be filled with liquid. The standard metric units tested are millilitres ($\text{ml}$), centilitres ($\text{cl}$), and litres ($\text{l}$):

$$\begin{aligned} 1\text{ cl} &= 10\text{ ml} \\ 1\text{ litre} &= 100\text{ cl} = 1{,}000\text{ ml} \end{aligned}$$

A crucial cross-domain relationship links metric capacity with cubic spatial volume:

$$1\text{ ml} = 1\text{ cm}^3 \quad \text{and} \quad 1\text{ litre} = 1{,}000\text{ cm}^3$$

This equivalence allows seamless translation between 3D shape dimensions and liquid capacities.

Metric Conversion Reference Table

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2. Imperial Units Tested in Context

Although the UK National Curriculum prioritizes metric units, 11+ entrance examinations frequently evaluate candidates' familiarity with everyday British imperial measures. Questions test both internal imperial conversions and approximate metric equivalents.

Imperial length and distance

Useful approximate benchmarks are 1 inch ≈ 2.5 cm, 1 pound ≈ 450 g and 1 imperial pint ≈ 570 ml. Use a conversion supplied by the question in preference to a remembered rounded value. The imperial pint and gallon here are UK units, not US liquid units.

• 1 foot ($1\text{ ft}$ or $1'$) = 12 inches ($12\text{ in}$ or $12''$)
• 1 yard ($1\text{ yd}$) = 3 feet = 36 inches
• 1 mile = 1,760 yards = 5,280 feet

The Metric-Imperial Distance Benchmark

One useful approximate conversion is between miles and kilometres:

$$5\text{ miles} \approx 8\text{ kilometres}$$

From this foundational ratio, we derive the unit rates:

$$\begin{aligned} 1\text{ mile} &\approx \frac{8}{5}\text{ km} = 1.6\text{ km} \\ 1\text{ km} &\approx \frac{5}{8}\text{ mile} = 0.625\text{ mile} \end{aligned}$$

To convert miles to kilometres, multiply by $8$ and divide by $5$ (or multiply by $1.6$). To convert kilometres to miles, multiply by $5$ and divide by $8$:

$$\text{Distance in km} \approx \text{Distance in miles} \times \frac{8}{5}$$ $$\text{Distance in miles} \approx \text{Distance in km} \times \frac{5}{8}$$

Rapid Benchmark Milestones:

• $10\text{ miles} \approx 16\text{ km}$
• $15\text{ miles} \approx 24\text{ km}$
• $25\text{ miles} \approx 40\text{ km}$
• $50\text{ miles} \approx 80\text{ km}$
• $65\text{ mph} \approx 104\text{ km/h}$

Imperial Capacity & Mass

• Pints and Gallons:
  • $1\text{ gallon} = 8\text{ pints}$
  • $1\text{ pint} \approx 568\text{ ml}$ (a pint is slightly more than half a litre)
  • $1\text{ gallon} \approx 4.5\text{ litres}$
• Pounds and Ounces:
  • $1\text{ pound (lb)} = 16\text{ ounces (oz)}$
  • $1\text{ stone} = 14\text{ pounds}$
  • $1\text{ kg} \approx 2.2\text{ lbs}$

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5. Reading Scales & Measuring Instruments

Questions involving measuring cylinders, thermometers, pressure gauges, dials, and weighing scales require determining the value of unlabelled graduation lines.

The Fundamental Scale Formula

To determine the numerical value of each interval division:

$$\text{Value per division} = \frac{\text{Upper marked value} - \text{Lower marked value}}{\text{Number of spaces (intervals) between marks}}$$

  100 ml ──┬──
           ├──  Interval 1
           ├──  Interval 2
           ├──  Interval 3
           ├──  Interval 4
   50 ml ──┴──  Interval 5

Calculation: (100 ml - 50 ml) ÷ 5 spaces = 10 ml per space

Reading Meniscuses & Negative Thermometer Scales

1. Liquid Meniscus: Surface tension curves liquid inside narrow cylinders. Read the volume at eye level from the bottom of the meniscus curve (concave surface for water).
2. Negative Scales: On thermometers reading below $0^\circ\text{C}$, remember that numbers increase in magnitude as you move downward (e.g., $-1^\circ\text{C}, -2^\circ\text{C}, -3^\circ\text{C}$). If each tick represents $2^\circ\text{C}$, two ticks below $0^\circ\text{C}$ is $-4^\circ\text{C}$, not $-2^\circ\text{C}$.

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Unit calculations

Worked Example 1: Multi-Step Metric Conversion & Material Usage

Problem: A carpenter has a timber plank measuring $3.6\text{ m}$ long. She cuts three identical shelves of length $75\text{ cm}$ each and two decorative strips of length $450\text{ mm}$ each. What length of the timber plank remains, in centimetres?

• Step 1 (Convert all dimensions to target unit, cm):
  • Initial plank length: $3.6\text{ m} = 3.6 \times 100 = 360\text{ cm}$.
  • Three shelves: each is $75\text{ cm}$. Total for shelves $= 3 \times 75 = 225\text{ cm}$.
  • Two strips: each is $450\text{ mm} = 450 \div 10 = 45\text{ cm}$. Total for strips $= 2 \times 45 = 90\text{ cm}$.
• Step 2 (Sum total wood cut): $$\text{Total cut} = 225\text{ cm} + 90\text{ cm} = 315\text{ cm}$$
• Step 3 (Calculate remaining wood): $$\text{Remaining length} = 360\text{ cm} - 315\text{ cm} = 45\text{ cm}$$
• Answer: $45\text{ cm}$.

Worked Example 3: Imperial-to-Metric Fuel & Distance

Problem: In this invented example, a motorist makes a 115-mile journey. The car uses 1 gallon of petrol for every 46 miles travelled. Petrol costs $\text{£}1.40$ per litre. Given that $1\text{ gallon} \approx 4.5\text{ litres}$, what is the approximate total fuel cost using that conversion?

• Step 1 (Calculate gallons used): $$\text{Gallons} = \frac{115\text{ miles}}{46\text{ miles per gallon}} = \frac{115}{46} = 2.5\text{ gallons}$$
• Step 2 (Convert gallons to litres): $$\text{Litres} = 2.5 \times 4.5 = 11.25\text{ litres}$$
• Step 3 (Calculate total fuel cost): $$\text{Cost} = 11.25 \times \text{£}1.40 = 11.25 \times 1.4 = \text{£}15.75$$
• Answer: approximately £15.75, using the supplied rounded gallon conversion.

Approximate imperial benchmarks checked against the UK government conversion table. The rounded classroom values are estimates, not exact conversion constants.