Units and conversions: the habit that saves marks

Ask a physics teacher where marks are lost and the answer is rarely "the idea was wrong". It is usually "the answer was right but in grams when the formula needs kilograms". Units are not decoration. They are part of the answer, and treating them carefully fixes a surprising number of mistakes.

Start with the SI base

Science agrees on one system so that formulas work without extra constants. The ones you meet most are the metre, the kilogram, the second, the ampere, the kelvin and the mole. Formulas like F = ma and V = IR assume you are in these units. Put 500 grams into F = ma and you will be out by a factor of a thousand.

A three-step conversion routine

1. Write the unit next to every number as you copy the question out. 250 mL, 3.5 h, 12 cm.
2. Convert to SI before substituting. 250 mL becomes 0.250 L, 3.5 h becomes 12,600 s, 12 cm becomes 0.12 m.
3. Carry units through the arithmetic. If distance in metres is divided by time in seconds, the result must be metres per second. If it is not, something is off.

Common conversions worth memorising

From	To	Do this
g	kg	divide by 1000
cm	m	divide by 100
mL	L	divide by 1000
cm³	m³	divide by 1,000,000
km/h	m/s	divide by 3.6
minutes	seconds	multiply by 60

The square and cube trap

Area and volume conversions trip up almost everyone. There are 100 cm in a metre, but 100 × 100 = 10,000 cm² in a square metre and 100 × 100 × 100 = 1,000,000 cm³ in a cubic metre. When you convert a squared or cubed unit, square or cube the conversion factor too.

Check the answer with dimensional analysis

Before you trust a result, check that the units on the right-hand side match the thing you wanted. Speed should come out as length over time. Density should be mass over volume. Power in watts is joules per second. If you ended up with metres squared where you needed metres, there is an extra multiplication hiding in your working.

A worked example

A block has mass 540 g and volume 200 cm³. For density in kg/m³: 540 g = 0.540 kg, and 200 cm³ = 0.000200 m³. Density = 0.540 ÷ 0.000200 = 2,700 kg/m³. Done directly in g and cm³ you get 2.7 g/cm³, and since 1 g/cm³ is 1000 kg/m³, both agree.