Mathematics II · Calculator allowed

Unit Conversions

Kilometres to metres, grams to kilograms, miles to kilometres, one simple method handles them all. Let's make it click.

A unit conversion just means writing the same amount in a different unit. The distance 3 km and 3000 m are the exact same distance, only the unit changed. The trick is knowing whether to multiply or divide, and by what.

Good news: there is one reliable method that works for every conversion, so you never have to guess the direction again. And since this is the calculator-allowed part of the test, the arithmetic itself is the easy bit. Let's walk through it.

The metric system: powers of 10

Metric (SI) units are built around multiples of 10, which is why they are so friendly. Each step on this staircase is a factor of 10. Going down a step (to a smaller unit) you multiply by 10; going up a step (to a bigger unit) you divide by 10.

kmhmdammdmcmmmsmaller unit: × 10 per stepbigger unit: ÷ 10 per step

Length is shown here, but the same staircase works for grams (mass) and litres (volume), just swap the base unit.

A handy conversion reference

These are the most common metric relationships. The Imperial / U.S. customary rates near the bottom are the kind of figures the test gives you, you will see the exact rate in the question, so you never have to memorize them.

QuantityRelationship
Length1 km = 1000 m
Length1 m = 100 cm
Length1 cm = 10 mm
Mass1 kg = 1000 g
Mass1 g = 1000 mg
Volume1 L = 1000 mL
Imperial (given)1 mile ≈ 1.6 km
Imperial (given)1 inch = 2.54 cm
Imperial (given)1 lb ≈ 0.45 kg
Good to know: the metric relationships above are worth memorizing because they are simple powers of 10. For Imperial / U.S. customary units, the test supplies the conversion rate inside the question, so focus on the method, not on remembering numbers like 2.54.

The one method: multiply by a conversion factor

A conversion factor is just a fraction that equals 1. For example, since 1 km = 1000 m, both of these fractions equal 1:

1000 m1 km = 11 km1000 m = 1

Multiplying by 1 never changes the amount, it only changes the unit. The skill is picking the version that cancels the unit you want to get rid of. Put that old unit on the bottom of the fraction so it cancels:

  • Write what you have with its unit, as a fraction over 1.
  • Multiply by the conversion factor arranged so the old unit cancels (old unit on the bottom, new unit on top).
  • Cancel and calculate, the old unit disappears, leaving the new unit and a number to punch into your calculator.

Worked example #1: 4.5 km to metres

We have kilometres and want metres. Since metres are smaller, we expect a bigger number. Use the factor that puts km on the bottom so it cancels.

4.5 km1 × 1000 m1 km(km on the bottom cancels)
= 4.5 × 1000 m = 4500 m
Answer: 4.5 km = 4500 m. The km units cancelled, leaving metres. And it makes sense: metres are smaller, so the number got bigger.

Worked example #2: 250 g to kilograms

This time we go from grams to bigger kilograms, so we expect a smaller number. We need the factor with grams on the bottom so the g cancels.

  • Have: 250 g.
  • Factor: 1 kg = 1000 g, so use 1 kg ⁄ 1000 g.
  • Calculate: 250 ÷ 1000 = 0.25.
250 g1 × 1 kg1000 g(g on the bottom cancels)
= 250 ÷ 1000 kg = 0.25 kg
Answer: 250 g = 0.25 kg. A kilogram is bigger than a gram, so 250 of the small units is only a fraction of one big unit.

Worked example #3: 12 miles to kilometres (rate provided)

Here is a typical Imperial question. The problem gives you the rate, say, 1 mile is about 1.6 km, so you do not need to remember it. The same cancelling method applies. We want miles to cancel, so miles go on the bottom.

12 miles1 × 1.6 km1 mile(miles cancel)
= 12 × 1.6 km = 19.2 km
Answer: 12 miles ≈ 19.2 km. Because each mile is longer than a kilometre, the number of kilometres comes out larger than the number of miles, a good sanity check.

Tips that keep you out of trouble

  • Predict the direction first. Going to a smaller unit gives a bigger number, and going to a bigger unit gives a smaller number. If your answer goes the wrong way, you flipped the factor.
  • Put the old unit on the bottom. Arrange the conversion factor so the unit you are leaving cancels. The unit you want should be left standing.
  • Use the rate the test gives you. For Imperial and U.S. customary units, the question supplies the conversion rate. Copy it down carefully and slot it into the method.
  • Keep the units in your working. Writing "km" and "m" right beside the numbers is the fastest way to catch a mistake before it happens.

Your turn: practice problems

A calculator is allowed, so the focus is on choosing the right factor. Predict whether the number should grow or shrink, then check yourself.

  1. Convert 3.2 L to millilitres (mL).
  2. Convert 750 cm to metres (m).
  3. Convert 2.5 kg to grams (g).
  4. Convert 5 inches to centimetres, using the given rate 1 inch = 2.54 cm.
Tap to reveal the answers
  • 1. 1 L = 1000 mL, and mL are smaller, so multiply: 3.2 × 1000 = 3200 mL.
  • 2. 1 m = 100 cm, and m are bigger, so divide: 750 ÷ 100 = 7.5 m.
  • 3. 1 kg = 1000 g, and g are smaller, so multiply: 2.5 × 1000 = 2500 g.
  • 4. Use the given rate 1 inch = 2.54 cm, with inches on the bottom: 5 × 2.54 = 12.7 cm.

Why this matters for the CAEC

Unit conversions show up all over the calculator-allowed part of the CAEC math test, in recipes, distances, weights, area and volume problems, and everyday measurement questions. The conversion-factor method is one reliable habit that handles every one of them, metric or Imperial.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.

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