EQN-02-10

Unit Conversion with Ratios

Chain conversion factors so the unwanted units cancel — the safe way to convert anything.

~5 min read•Difficulty 2/5

Because $1 m = 100$ cm, the ratio (100 cm / 1 m) equals 1. Multiplying by it changes the units but not the amount.

Write the factor so the unit you want to remove sits opposite where it currently is, then cancel.

250 cm \times (1 m /100 cm) = 2.5 m (cm cancels)

2 hours \to seconds :

2 h \times (60 min /1 h) \times (60 s /1 min) = 7200 s

Area and volume conversions square or cube the factor.

1 m = 100 cm, so 1 m^{2} = 10 0^{2} = 10000 cm^{2}, and 1 m^{3} = 10 0^{3} = 1000000 cm^{3} .

Exchange rates are just conversion factors. Compose them the same way: cancel the middle currency.

Worked examples

Example 1 (difficulty 1/5)

Convert 3.5 kilometres to metres.

◎ Predict the answer first:

Example 2 (difficulty 2/5)

How many square centimetres are in 2 m²?

◎ Predict the answer first:

Write each factor so the unwanted unit cancels — let the units guide the arithmetic.
Square the factor for areas, cube it for volumes.
Going to a smaller unit multiplies; to a larger unit divides.

Using the linear factor for area or volume

Why: It is the factor you remember.

Fix: Square it for area

(100 \to 10000)

and cube it for volume.

Multiplying when you should divide (or vice versa)

Why: The direction is guessed.

Fix: Set up the factor so units cancel; the arithmetic direction follows automatically.