A correct method can still lose marks when the units are mixed. Convert every quantity to the same units before calculating, then check that the size of the answer makes sense.
This lesson is part of SPM subjects and learning resources. It goes well with supporting a design explanation with evidence, which uses the same kinds of numbers.
What is the unit-first routine?
Follow four steps every time.
- List each given quantity with its unit.
- Decide the unit the answer needs.
- Convert the given quantities to that unit system.
- Calculate, then compare the answer with a size you know.
The routine takes under a minute. It stops the most common slip, which is mixing mm and m in one line.
Worked example: concrete for a small slab
An original classroom problem: a concrete slab is 4.0 m long, 3.0 m wide and 150 mm thick. Concrete has a density of 2 400 kg/m³. Find the volume and the mass.
Step 1: list. Length 4.0 m, width 3.0 m, thickness 150 mm, density 2 400 kg/m³.
Step 2: target. Volume in m³, mass in kg.
Step 3: convert. Thickness: 150 mm ÷ 1 000 = 0.15 m.
Step 4: calculate. Volume = 4.0 × 3.0 × 0.15 = 1.8 m³. Mass = 2 400 × 1.8 = 4 320 kg.
Sense check: a slab about the size of a small room, thinner than a hand is wide, holds under two cubic metres. A mass of about four tonnes is reasonable for that amount of concrete.
The mistake that costs marks
The usual slip is to put 150 straight into the multiplication. The working looks complete, but the answer is 1 000 times too large.
| Step | Wrong | Right |
|---|---|---|
| Thickness used | 150 (mm left as is) | 0.15 m |
| Volume | 4.0 × 3.0 × 150 = 1 800 m³ | 4.0 × 3.0 × 0.15 = 1.8 m³ |
| Sense check | Not done | Flags 1 800 m³ as far too big |
A second check is to read the units aloud: m × m × mm is not m³. If the units do not combine to the unit you need, stop and convert.
Converting areas and volumes
Convert lengths before multiplying. Converting afterwards needs a factor that is easy to get wrong.
| Quantity | Wrong shortcut | Safe method |
|---|---|---|
| 600 mm × 400 mm in m² | 600 × 400 ÷ 1 000 | 0.6 × 0.4 = 0.24 m² |
| 2 500 mm × 300 mm × 300 mm in m³ | ÷ 1 000 | 2.5 × 0.3 × 0.3 = 0.225 m³ |
Converting each length first uses a conversion you already know. The arithmetic is slightly longer, but the answer is far more reliable.
Check yourself
A concrete column is 3.0 m high with a square cross-section of 300 mm by 300 mm. Find its volume in m³ and its mass if the density is 2 400 kg/m³.
Answer
Convert first: 300 mm = 0.30 m.
Volume = 0.30 × 0.30 × 3.0 = 0.27 m³.
Mass = 2 400 × 0.27 = 648 kg.
Sense check: a column about as high as a room and as thick as a large book is a little over a quarter of a cubic metre, so 0.27 m³ is reasonable.
What to study next
Go on to explaining a concept with a clear sequence, or test yourself on the original mixed practice. The structured answer self-review tool helps you check units line by line.
For a teacher to go through your calculations with you, see online one-to-one Civil Engineering Studies tuition.