MSC #9: Proportionately – Inverse Proportion

In our previous post, MSC #8, we discussed direct proportion, that is when a quantity increases the other quantity also increases. y is said to be directly proportional to x if y/x = k where k is the constant of proportionality. Thus, we applied this principle in reducing fractions to their “lowest terms” and simplifying division problems.

In inverse proportion when one value increases, the other value decreases. y is said to be inversely proportional to x if y = k/x or xy = k, where k is the constant of proportionality. Our Master strategy for today is to use this concept to simplify multiplication problems including its application in percentage calculations.

One of the simplest applications of inverse proportion is our technique of  “doubling and halving together”:

4 x 18 is easier computed by doubling 4 while halving 18 leading to 8 x 9 = 72.

5 x 64 is easily 10 x 32 or 320 while 35 x 16 can be quickly computed as 70 x 8 or 560.

It is also very helpful in multiplication involving decimals. 3500 x 0.02 can be easily transformed in an easy 35 x 2 = 70 multiplication by simply moving the decimal point of 0.02 two places to the right while moving the decimal point of 3,500 the same number of decimal places in the opposite direction.

That procedure can be extended to percentage computations, such as 84% of 25. The “of” here, suggest multiplication and since percent means per hundred, that expression is equivalent to 0.84 x 25. By shifting decimal points, the multiplication can be transformed into an easier 84 x 0.25.

Hence, 84% of 25 = 0.84 x 25 = 84 x 0.25 = 0.25 x 84 = 25% of 84.

Now 25% is equivalent to 1/4 and 1/4 of 84 is 21.

Also, we have, 48 % of 75 = 75% of 48 = 3/4 x 48 = 36

Our featured example, 36% of 25 + 75% of 36, aside from being evaluated as shown in the figure, can be also simplified as:

36% of 25 + 75% of 36

= (0.36 x 25) + (0.75 x 36)

= (0.36 x 25) + (75 x 0.36)

= (0.36 x 100) = (36 x 1) = 36