Quarter Wit, Quarter Wisdom: Varying Directly

Quarter Wit, Quarter WisdomWe can keep working on ‘pattern recognition’ questions for a long time and not run out of questions of different types on which it can be used. We hope you have understood the basic concepts involved. So let’s move on to another topic now: Variation.

Basically, variation describes the relation between two or more quantities. e.g. workers and work done, children and noise, entrepreneurs and start ups. More workers means more work done; more children means more noise; more entrepreneurs means more start ups and so on… These are examples of direct variation i.e. if one quantity increases, the other increases proportionally. Then there are quantities that have inverse variation between them e.g. workers and time taken. If there are more workers, time taken to complete a work will be less.

Let’s discuss direct variation today.

Formally, let’s say x varies directly with y. If x takes values x1, x2, x3… and y takes values y1, y2, y3 … correspondingly, then x1/y1 = x2/y2 = x3/y3 = Some constant value

In other words, ratio of x and y stays the same in different instances.

(Notice that this is the same as x1/x2 = y1/y2)

It might seem a little cumbersome when put this way but the truth is that direct variation is quite intuitive. A couple of questions will make it clear.

Question 1: 20 workmen can make 35 widgets in 5 days. How many workmen are needed to make 105 widgets in 5 days?

(A)   7

(B)   20

(C)   25

(D)   30

(E)    60

Solution: Notice that the number of days stays the same so we can ignore it. Now think, how are workmen and widgets related? If the number of workmen increases, the number of widget made also increases proportionally. You need to find the new number of workmen required. The number of widgets has become thrice (105/35 = 3) so number of workmen needed will become thrice as well (remember, the number of workmen will increase in the same proportion).

We need 20*3 = 60 workmen

Answer (E)

The concept of variation is very intuitive. If the number of widgets required doubles, the number of workmen required to make them in the same amount of time will double too. If the number of widgets required becomes one fourth, the number of workmen required to make them in the same amount of time will become one fourth too.

A quantity can directly vary with some power of another quantity. Let’s take an example of this scenario too.

Question 2: If the ratio of the volumes of two right circular cylinders is given by 64/9, what is the ratio of their radii? (Both the cylinders have the same height)

(A)   4/3

(B)   8/3

(C)   16/9

(D)   4/1

(E)    16/3

Solution: This question involves a little bit of geometry too. The volume of a right circular cylinder is given by Area of base * height i.e.

Volume of a right circular cylinder = pi*radius^2 * height

So volume varies directly with the square of radius.

Va/Vb = 64/9 = Ra^2/Rb^2

Ra/Rb = 8/3

Answer (B)

We hope this little concept is not hard to understand. We will work on inverse proportion next week and then work on problems involving both (that’s where the good questions are!).

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

6 Responses

  1. Lavanya says:

    Hi Karishma,

    What about height?
    How are we deducing this -” So volume varies directly with the square of radius.”?


    • Karishma says:

      When dealing with variation, our assumption is that if ‘a’ varies with ‘b’, any other variable that ‘a’ varies with stays constant. Volume varies with the square of the radius (keeping height constant)
      a/b = Constant
      V/r^2 = Constant
      I am mentioning specifically that height is constant in the question (since it seems to be creating confusion) but it is assumed that other than the quantities we are considering, everything is constant.

  2. Lavanya says:


    Can you please take up number line in one of your posts? I am struggling with number line and it would be great if you could explain this concept.


    • Karishma says:

      Sure but I don’t really know what you want me to explain. The number line is a hypothetical one dimension line on which all real numbers lie…

      …………-4..-3..-2.5…-2……0..0.1…. 0.4……….3…4…5…and so on

      We use the number line to solve different types of questions e.g. mods, SD, weighted avgs, number properties etc
      What is giving you trouble?

      • Lavanya says:

        Hi Karishma,

        I am having difficulty applying this concept to mods and number properties in particular. It would be terrific If you could explain the concept itself and its various applications

        Thank you so much for your posts. These are incredibly helpful and actually train you to think more and write less :)


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