Foundation lesson
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
- A steady walk covers 14 metres in 10 seconds.
- 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.
- 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.
- Metres and seconds are different units, so neither cancels and what comes out is a new quantity that keeps a unit of its own.
- 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.
One in four and two in eight, drawn side by side
The filled dots are the ones that moved. Under each row, a bar with that share of its length shaded. The two groups have different sizes and the two shaded stretches are the same length.
The shaded stretch is the same length in both bars. A label that long does not fit inside a drawing, and a drawing's text does not reflow on a narrow screen, so it is written here.
What it shows, in words
The first row is a group of 4 with 1 filled, and the bar under it is shaded for a quarter of its length. The second row is a group of 8 with 2 filled, and the bar under it is shaded for the same length as the first. Doubling how many moved and doubling the size of the group together leave the share where it was, at 0.25.
The dots are counted and the bars are measured, so the two rows agree by two different routes. A fraction records how much of the group moved. The size of the group is a separate fact, and the fraction has already divided it out.
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.
If you came here from a passage, Back returns you to the exact place you left.