# Fractions and ratios

A fraction a/b says how many times b fits into a. Multiplying the top and the bottom by the same number leaves it unchanged.

Written for this edition, not translated from Einstein.

## What does dividing one quantity by another tell you?

The fraction 6/2 is 3, because 2 fits into 6 three times; it also says that 6 is three times 2. Every fraction a/b works the same way. The bottom number can never be zero, since no count of zeros adds up to 6.

When the two quantities have different units, their ratio is a new quantity with a unit of its own: 12 micrometres travelled in 4 seconds is 3 micrometres per second. When they have the same unit, the units cancel and the ratio is a plain number: 2 particles out of 8 is 0.25 of the group.

## Worked example: Fourteen metres in ten seconds

1. A steady walk covers 14 metres in 10 seconds.
2. Divide the distance by the time: 14 divided by 10 is 1.4, and the unit divides with it, so the rate is 1.4 metres per second.
3. After 60 seconds the same walk has covered 84 metres, and 84 divided by 60 is 1.4 again. The walk is steady, so the ratio does not depend on how long it is watched.
4. Metres and seconds are different units, so neither cancels and what comes out is a new quantity that keeps a unit of its own.
5. The drawing below is the other case: 2 particles out of 8 has the same unit above and below, they cancel, and 0.25 is left as a plain number.

## Where this lesson stops

A ratio of two quantities in the same unit is a plain number. A ratio of different units, such as micrometres per second, is a new quantity and keeps its unit.

## This lesson builds on

- [Adding and averaging](/foundations/bridge-sum-average/)