Multiplication and division structures

The ratio structure for division

The ratio structure for division refers to situations where we use division to compare two quantities. In a situation where a comparison has to be made between two numbers, two sums of money or two measurements of some kind, we can make the comparison by subtraction, focusing on the difference between the quantities.

For example, if A earns €300 a week and B earns €900 a week, one way of comparing them is to state that B earns €600 more than A, or A earns €600 less than B. The 600 is arrived at by the subtraction, 900 – 300. But we could also compare A’s and B’s earnings by looking at their ratio, stating, for example, that B earns three times more than A. The three is obtained by the division, 900 divided by 300. This process is simply the inverse of the scaling structure of multiplication described above, since what we are doing here is finding the scale factor by which one quantity must be increased in order to match the other. In the above example, the question would be, by what factor must 300 be multiplied to give 900?