MSC #9 – Proportionately II: Inverse Proportion

Yesterday we discussed direct proportion, that is when a quantity increases by a certain percentage the other quantity increases by the same percentage. It is 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 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. We can use this concept to simplify multiplication problems.

The simplest form is “doubling and halving together:

4 x 18 is easier computed 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.

Our featured example shows how the shortcut for multiplying by 25 was developed.  We can apply “doubling and halving” together twice to get 848 x 25 = 424 x 50 = 212 x 100 = 21,200. The short cut then is “when multiplying a number by 25, divide the number by 4 and move the decimal point two places to the right”.

AS we have written yesterday, doubling and Halving will be discussed by IAVM co-founder Swati Dave and Proportionately by Rolito Asombra, LET board topnotcher and VM coach of the San Jose National High school, San Pablo City in the Inspirational Math from India webinars on Dec 5, 2021.

This technique is very useful in percentage calculations. Since percent means per hundred we can multiply the percentage figure by 100 while at the same time dividing the other figure by 100.

So we can transform 16% of 25 to 25% of 16 and 48% of 75 into 75% of 48 and make calculations easier.

16% of 25 = 0.16 x 25 = 16 x 0.25 = 0.25 x 16 = 25% of 16 = 4

48 % of 75 = 75% of 48 = 3/4 x 48 = 36

We can use the above technique in combination with number splitting:

72% of 17.5 = 17.5% of 72 = 10% of 72 + 5% of 72  + 2.5% of 72 = 7.2 + 3.6 + 1.8 = 12.6

 Doubling and halving is also discussed in Chapter 6 of the Inspirational Maths from India: a Teacher’s Handbook by IAVM while Direct and Inverse Proportions are the subjects of Chapters 8 and 9 of the 30 Master Strategies in Computing by Veronica and Virgilio Prudente.