Factor Change

When a quantity gets larger or smaller, we say that it changes.

For example, the volume of a certain liquid in a container can change from 3 quarts to 12 quarts. One could simply imagine extra liquid being poured into the container taking the volume from 3 quarts to 12 quarts.

3 quarts is changed to 12 quarts


Factor changes are not additions or subtractions.

We can talk about this change in volume in several ways. One way is to say that 9 quarts were added, since 3 quarts plus 9 quarts is equal to 12 quarts. Talking about the change this way, however, is not talking about it in terms of a factor change. This is talking about the change as an additive change, not a factor change.

An additive change


Factor changes are multiplications.

To understand factor change, think about it this way: 3 times 4 equals 12. So, times 4 is the multiplier that changes the 3 quarts into 12 quarts. We say, therefore, that the volume changed by a factor of 4.

A factor change

As we shall see next, the above mathematics shows a factor change of 4.


Factor changes are always discussed in terms of the multiplier that takes you from the initial quantity to the final, or changed, quantity.

A factor change of three (3):

If a distance changed from 5 meters to 15 meters, we would say that there was a factor change of 3 in the distance, since 5 times 3 is 15. We could also say, 'The distance changed by a factor of 3'.

A factor change of one half (1/2):

If a speed changed from 40 m/s to 20 m/s, we would say that the speed changed by a factor of 1/2. This is because 40 times 1/2 equals 20.


Again, the factor change in a quantity is the multiplier that takes you from the initial value of some quantity to the final, or changed, value of that quantity.

Here are some more examples, read across:

Initial Quantity Final Quantity Factor Change Since
3 m 30 m 10 (3 m)(10) = 30 m
40 s 10 s 1/4 or 0.25 (40 s)(1/4) = 10 s
4.6 m/s 12.8 m/s 12.8/4.6 or about 2.78 (4.6 m/s)(12.8/4.6) = 12.8 m/s
5.3 kg 1.1 kg 1.1/5.3 or about 0.21 (5.3 kg)(1.1/5.3) = 1.1 kg

Notice that the factor change in a quantity can easily be calculated by taking the final quantity and dividing it by the initial quantity. For example:

  • If a quantity starts out with a value of 7.2 (initially)
  • And that quantity changes to a value of 9.6 (finally)
  • Then the factor change is (9.6 / 7.2), or about 1.3
  • Because 7.2 times (9.6 / 7.2) equals 9.6


Proportions can be defined in terms of factor changes.

Factor changes are important to understand when discussing proportions. Here is some related material:

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