Quantities & units

The milk carton says 1 l, the bag of flour says 1 kg, the ruler is marked in centimetres and the road sign gives the distance in kilometres. Whenever you measure something, two things come together: a number and a unit. Once you keep the two apart cleanly, you can convert, compare and check any measurement yourself. That is exactly what you practise here.

Number and unit belong together

A quantity is anything you can measure: a length, a mass, a volume, a time. The unit says which piece you measure with, the number says how many of those pieces there are. So 3 m means: three times the piece called metre. That is why the statement 3 on its own says nothing at all, because 3 g, 3 l and 3 km are completely different things. And you can only add up measurements of the same quantity: 2 kg plus 500 g makes sense, 2 kg plus 500 ml does not.

Milli, centi, kilo

To keep numbers handy, prefixes are put in front of the unit. Milli means thousandth: 1 mm is a thousandth of a metre, 1 ml is a thousandth of a litre. Centi means hundredth: 1 cm is a hundredth of a metre. Kilo means thousand: 1 km is 1000 m, 1 kg is 1000 g. So a prefix is nothing but a hidden number in front of the unit. For large masses there is also the tonne, 1 t is 1000 kg. Only time refuses to join in: there the steps go in sixties, because 1 min is 60 s and 1 h is 60 min. That is why we convert only lengths, masses and volumes here.

Smaller unit, bigger number

Converting only changes how you write a measurement, the measured quantity stays the same. If you switch to a smaller unit you need more pieces, so the number grows and you multiply: 3 m is $3 \cdot 100 = 300$ cm. If you switch to a bigger unit you need fewer pieces, so the number shrinks and you divide: 4500 g is $4500 : 1000 = 4.5$ kg. What you multiply or divide by is the conversion factor between the two units. And because that is always 10, 100 or 1000, all that really happens is the decimal point moving: times 100 pushes it two places right, divided by 1000 three places left.

In everyday life you jump between these steps all the time. A recipe asks for 250 g of flour, the pack holds 1 kg, that is 1000 g, and $1000 : 250 = 4$ tells you it is enough for four batches. A bottle holds 1.5 l and a glass holds 200 ml, and since 1.5 l is exactly 1500 ml, $1500 : 200 = 7.5$ gives you seven and a half glasses. The safe route is always the same: first bring all measurements into the same unit, then calculate, and at the end switch back to the unit the question asked for. For really large masses the tonne helps, a small car weighs about 1.2 t, that is 1200 kg.