Multiplicative Thinking 2

Constant Product situations

In these situations, when one of the parameters changes in its value, the other parameter changes in such a way that their product always remains same. When one of the parameters increases, the other decreases and vice versa.

A few examples will clarify this idea.

Painting a house

Imagine that you have given a contract to a company to paint your house. The contractor says that the work will take 10 days and he deputes 2 painters to do the work. Suppose now you want the painting to be completed in 5 days and inform the contractor. What will he do? It is obvious that he would have to send more painters to work. So, you can see that when there are more painters, the painting would get done in a shorter time and vice versa.

Under certain circumstances we would see that the number of painters X number of days needed to complete painting can be seen as a quantity which remains same. We can even think of it as a number which represents the total amount of work!

If it would have taken 10 days for 2 painters to complete the work, we could say that the total amount of painting work is 20 painter days! Assuming that all painters work with the same efficiency, we can say that this work of 20 painter days will remain same. Hence if we want the painting to be done in 5 days then the number of painters required will be “20 painter days/ 5 days” which is equal to 4 painters.

Here the total work was represented as “20 painter days”. This is not a standard unit of measurement, but a unit which we have created for our convenience, in a particular context.

Constant Ratio & Constant Product Situations

We know that like addition & subtraction, multiplication & division are also related. What is multiplication in one perspective can be seen as division from another perspective. Similarly, both the above situations occur in the same situation, depending on our perspective.

Shopping

In the previous chapter we saw a constant ratio situation in shopping. Now if you assume that the total amount of money that I have is constant, the more costly the product the less quantity I would be able to purchase. Hence here the price and quantity have a constant product relations!

Travel

If the total distance we are travelling is same, then higher the speed of the car, the less time it would take to travel the distance. Hence the speed and the time taken have a constant product relationship.

Inverse variation

Constant Product situations are also called “inverse proportion” situations, because both the units under consideration change in opposite directions.

One of the famous inverse relations is in economics which says that when the supply of a thing reduces, its price would increase!

In real life, many pairs of parameters may change in opposite ways but their product may not remain constant. For example, in real life all painters may not work at the same efficiency and some part of the house may take more effort to paint than other parts. For example painting a wall would be easier that painting a kitchen though the surface area may be same. In primary school we study only such situations where the product is constant.