Math of Proportions

Rate Approach or Unitary Method

The underlying assumption is that the “price” or “rate” in terms of Rs per Kg remains same in both transactions.

First, we look at the table horizontally and see that 5 kgs of vegetables cost Rs 80. Hence the rate at which they are sold is given by dividing the rupee amount 80 by the quantity 5. The result is Rs 16 per Kg.

Next, we apply the same price to 8 kgs and the amount payable is got by multiplying the rate 16 by the quantity 8. The answer if Rs 128.

This approach is traditionally called the “Unitary Method”.

Ratio or Scale Method

There is another way of solving the problem by looking at the “scale factors” involved.

First, we look at the table vertically and see that the quantity changes from 5 to 8 from the first scenario to the next. We can say that the ratio increases in the ratio 5:8.

Here again the assumption is that the “price” does not change. This implies that the amount to be paid also increases in the ratio 5:8. So -

cost 1 : cost 2 :: 5 : 8.

Hence cost 2 = 8/5 X 80 which is Rs 128.

Quantities in Proportion

The essential math relation in both cases can be summarized as below. When 2 ratios a:b & c:d are equal (in terms of their fractional value), they are said to be in proportion.

Hence a/b = c/d. This means a X d = b X c or ad = bc.

It can also be written as a/c = b/d.

In a typical problem, both the ratio equalities would be such that one of them would be a rate approach and the other would be a scale approach. So, both rate & scale approaches to a problem are just 2 different ways of looking at the same problem.