Proportion

The world of mathematics of numbers is vast and varied. When your child was introduced to numbers, their main focus was on understanding value and place. Then the focus shifted to internalising operations such as addition, subtraction, multiplication and division. From there they journeyed through the lands of factors, multiples and prime numbers. Now, they have arrived here, in the domain of proportions.

The Big Idea: What are Proportions?

These three topics are a natural extension of the concept of fractions and decimals. Therefore, make sure you check out those pages and thoroughly understand them before attempting to start on proportions.

The basics of proportion

When two ratios are equivalent, they are said to be in proportion. For example, \(2 : 5\) and \(10 : 25\) are equivalent ratios. This means they are in proportion. We represent this by writing \(2 : 5 : : 10 : 25\) and read this as ‘2 is to 5 as 10 is to 2.

Procedure - Direct variation

When the ratio of the two quantities is constant, the quantities are in direct variation. Increase in one quantity causes an increase in the other and decrease in one quantity causes decrease in the other. Let’s try and understand this using an example,

Example: The price of a pen is ₹12. The table below shows the number of pens bought by Dev and their price.

Number of pens (N)	1	2	3	4	5
Total price (P)	12	24	36	48	60
\(\begin{align}N\over P \end{align}\)	\(\begin{align}1\over 12 \end{align}\)	\(\begin{align}{1\over 12}={2\over 24} \end{align}\)	\(\begin{align}{1\over 12}={3\over 36} \end{align}\)	\(\begin{align}{1\over 12}={4\over 48} \end{align}\)	\(\begin{align}{1\over 12}={5\over 60} \end{align}\)

As you would notice that there is an increase or decrease in price as there is an increase and decrease in the number of pens. The ratio of the number of pens to the total price is always the same. We can see that \(1 : 12 = 2 : 24 = 3 : 36\) and so on. \(N\) and \(P\) in direct variation. 

This direct variation is written as \(P∝ N,\) read as \(“P\) is directly proportional to \(N”\) or \(“P\) varies directly as \(N”.\) When two quantities, \(a\) and \(b,\) are directly proportional, we can say that \(a ∝ b\) and \(b ∝ a.\)

Procedure - Inverse variation

When the ratio of one quantity to the reciprocal of the other quantity is constant, the two quantities are in inverse variation.

Number of men (N)	1	2	3	6	8
Number of days (D)	48	24	16	8	6
\(\begin{align}N\over D \end{align}\)	\(\begin{align}{1\over{1\over48}}=48\end{align}\)	\(\begin{align}{2\over{1\over24}}=48\end{align}\)	\(\begin{align}{3\over{1\over16}}=48\end{align}\)	\(\begin{align}{6\over{1\over8}}=48\end{align}\)	\(\begin{align}{8\over{1\over6}}=48\end{align}\)

As you can observe with the increase in the number of men, the number of days required to complete the work decreases. The ratio of the number of men and the reciprocal of the number of days is constant.

We can see that \(\begin{align}1:{1\over48}=48, \;2:{1\over24}=48, \;3:{1\over16}\end{align}\)

This is written as \(\begin{align}N ∝ {1\over D},\end{align}\) read as “N is inversely proportional to D” or “N varies inversely as D”

When two quantities, \(a\) and \(b,\) are inversely proportional, we can say that \(\begin{align}a ∝ {1\over b} \text { and } b ∝ {1\over a}∝ 1 \end{align}\)

More the number of men, the lesser the number of days required to finish the work.

Constant of proportionality

Time of travel (T) (Hr)	1	2	3	4	5
Distance covered (D) (Km)	50	100	150	200	250
\(\begin{align}{D\over T}{(Km/hr)} \end{align}\)	50	50	50	50	50

As you can see \(D ∝ T,\) \(DT\) is a constant and its value is called the ‘constant of proportionality’. In general, when \(a\) varies directly as \(b,\) we write \(\begin{align}{a\over b} = k\;\;(\text{or}) \;\;a= kb \end{align}\), where ‘k’ is called the constant of proportionality. 

 

Tips and Tricks

• Proportions can be direct or indirect. A trick is to always think of the situation from real life and determine whether we are dealing with direct or inverse proportions.
  E.g. 10 people finish a task in a certain amount of time. How much time will it take 20 people?
  At first glance, seeing 10 becoming double raises the temptation of doubling the time. But if you think of the situation, you’ll recognize that more people will get the same work done is a shorter amount of time. So this is a case of inverse variation.
  Build your intuition this way so that you follow the correct procedure.

Test your knowledge 

Q1. Answer the following question

• The ages of Moni and Nivi are in the ratio \(2 : 3.\) Ten years from now, their ages will be in the ratio \(4 : 5.\) What are their present ages?