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Ratios and Proportions


The quantitative aptitude section in any competitive exam has a variety of questions. One of them is Ratio and proportion. Along with ratio and proportion, there is one more important topic that you need to prepare and it is variation. The question on this topic comes in terms of variation in ratios. Variation is a very simple concept.

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Direct Variation

Variation is a very simple concept. If two quantities are dependent on each other, one will vary according to the other. There are many different types of variation in ratios and proportion and they are:

variation in ratios

In this, a quantity ‘x’ is said to vary directly with ‘y’ or directly proportional to ‘y’, if ‘x’ increases or decreases respectively by as many times as ‘y’ increases or decreases. The ratio here which is x:y is constant.


p α q, i.e. p = k × b, where k is constant.
It follows that p/q = k. If two sets of data are given in the question then we can write that p1/q1 = p2/q2.

Inverse Variation

This is another type of variation in ratio and proportion. A quantity ‘x’ is inversely or be inversely proportional to ‘y’ if ‘x’ respectively increases or decreases by as many times as ‘y’ decreases or increases. Here, the important thing to note is that if x increases then y decreases and if x decreases then y increases. This is known as inversely proportional to each other. The p×q is constant here. This is expressed as below:

p α1/q, i.e. p = k/q, where k = constant.
It follows that px q = k. And if there are two sets of data given to you then, p1 x q1 = p2 x q2

Joint Variation

This is the third and final type of variation. A quantity is said to vary jointly with two or more other quantities if it varies with each quantity, while the remaining quantities being constant. p varies jointly with q and r if it varies with q when r is constant and varies with r when q is constant.

For example, p α qr => p = kqr, where k is constant.

Example of Variation in Ratios

Q. The ratio of yellow balls to green in a bag P is the same as that of red to black in bag Q. This ratio is 9:5. The red and green balls are exchanged between the bags i.e bag P now was yellow and red balls while bag Q has red and green balls. Find the ratio of total balls in bad P and bag Q.

A. 1              B. 14:5                C. 9:5                    D. Data Not Sufficient

Ans: In the given question the ratio of yellow/green and red/black is 9/5. And the red and green balls are exchanged and we are required to find the total number of balls in each bag. Assume that the total balls in each bag are 14, with bag P containing 9 yellow and 5 green balls and similarly bag Q containing 9 red and 5 black balls.

Now, that we exchange the red and green balls, bag P will 9 yellow balls and 9 red balls. Similarly the other bag, bag Q will 5 green balls and 5 black balls. So, the required ratio of the total balls in both the bags is 18:10 or 9:5. So, the correct answer is C.

Practice Questions

1. a is the mean proportional of (a + 2) and (a – 6). Find A.

A. 3             B. -3                C. -6                 D. Data Not Sufficient

Ans. The correct answer is B.

2. A room contains boys and girls. If there are twice as many boys as girls, find the ratio of the girls to the total people in the room.

A. 1:3            B. 1:6                C. 1:2                   D. 1:1

Ans. The correct answer is A.

3. The number of male teachers in a school is 20% more than a number of female teachers in the school. What is the ratio of male and female teachers?

A. 5:4            B. 6:5               C. 5:6                  D. Data Not Sufficient

Ans. The correct answer is C.

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