Converting Measurement Units

The same amount, written differently

$$5 \text{ meters} = 500 \text{ centimeters}$$

Nothing changed except the unit. The length is identical — a centimeter is smaller, so it takes more of them to cover the same distance.

That is the whole idea, and it decides every conversion.

Multiply or divide

Going to a	The number	So you
smaller unit	grows	multiply
bigger unit	shrinks	divide

$$5 \text{ m} \rightarrow 500 \text{ cm} \qquad \text{(smaller unit, multiply)}$$ $$300 \text{ cm} \rightarrow 3 \text{ m} \qquad \text{(bigger unit, divide)}$$

The metric system runs on tens

Every metric step is a factor of $10$, $100$ or $1000$, and the prefix names it.

Prefix	Means
kilo-	one thousand
centi-	one hundredth
milli-	one thousandth

Conversion	Factor
km → m	$\times 1000$
m → cm	$\times 100$
cm → mm	$\times 10$
m → km	$\div 1000$
cm → m	$\div 100$

The same prefixes work for mass and capacity. A kilogram is a thousand grams; a milliliter is a thousandth of a liter. Learning the prefixes once covers every metric unit.

Because the factors are powers of ten, converting only moves the decimal point:

$$2.5 \text{ km} = 2500 \text{ m} \qquad 45 \text{ mm} = 4.5 \text{ cm}$$

Customary units

These have no single pattern, so the factors have to be known.

Conversion	Factor
feet → inches	$\times 12$
yards → feet	$\times 3$
miles → feet	$\times 5280$
pounds → ounces	$\times 16$
gallons → quarts	$\times 4$

The method is unchanged. Only the numbers are less tidy.

$$4 \text{ feet} = 4 \times 12 = 48 \text{ inches}$$ $$60 \text{ inches} = 60 \div 12 = 5 \text{ feet}$$

Convert before calculating

A problem with mixed units has to be brought to one unit first.

A rope is $2$ m long. Another is $150$ cm. What is the total?

Adding $2$ and $150$ gives $152$ of nothing. Convert first:

$$2 \text{ m} = 200 \text{ cm}$$ $$200 + 150 = 350 \text{ cm}$$

Or in meters: $2 + 1.5 = 3.5$ m. Either unit works — what fails is using both at once.

Choose whichever unit avoids awkward numbers. Here centimeters keeps everything whole.

Why the direction check is worth doing

Almost every error in this topic is multiplying where you should divide, and it is always caught by asking whether the answer is sensible.

$$300 \text{ cm} = 3 \text{ m}$$

If you had multiplied instead, you would get $30{,}000$ m — thirty kilometers. A rope thirty kilometers long is not a plausible answer to a question about a rope, and no arithmetic check is needed to see it.

That habit is worth more than memorizing which conversions multiply. The unit tells you the direction, and the direction tells you the operation.

Worked examples

Common mistakes

Practice problems