Now that we have discussed direct and inverse variation, joint variation will be quite intuitive. We use joint variation when a variable varies with (is proportional to) two or more variables.

Say, x varies directly with y and inversely with z. If y doubles and z becomes half, what happens to x?

“x varies directly with y” implies x/y = k (keeping z constant)

If y doubles, x doubles too.

“x varies inversely with z” implies xz = k (now keeping y constant)

If z becomes half, x doubles.

So the overall effect is that x becomes four times of its initial value.

The joint variation expression in this case will be xz/y = k. Notice that when z is constant, x/y = k and when y is constant, xz = k; hence both conditions are being met. Once you get the expression, it’s very simple to solve for any given conditions.

x1*z1/y1 = x2*z2/y2 = k (In any two instances, xz/y must remain the same)

x1*z1/y1 = x2*(1/2)z1/2*y1

x2 = 4*x1

Let’s look at some more examples. How will you write the joint variation expression in the following cases?

1. x varies directly with y and directly with z.

2. x varies directly with y and y varies inversely with z.

3. x varies inversely with y^2 and inversely with z^3.

4. x varies directly with y^2 and y varies directly with z.

5. x varies directly with y^2, y varies inversely with z and z varies directly with p^3.

Solution: Note that the expression has to satisfy all the conditions.

1. x varies directly with y and directly with z.

x/y = k

x/z = k

Joint variation: x/yz = k

2. x varies directly with y and y varies inversely with z.

x/y = k

yz = k

Joint variation: x/yz = k

3. x varies inversely with y^2 and inversely with z^3.

x*y^2 = k

x*z^3 = k

Joint variation: x*y^2*z^3 = k

4. x varies directly with y^2 and y varies directly with z.

x/y^2 = k

y/z = k which implies that y^2/z^2 = k

Joint variation: x*z^2/y^2 = k

5. x varies directly with y^2, y varies inversely with z and z varies directly with p^3.

x/y^2 = k

yz = k which implies y^2*z^2 = k

z/p^3 = k which implies z^2/p^6 = k

Joint variation: (x*p^6)/(y^2*z^2) = k

Let’s take a GMAT prep question now to see these concepts in action:

Question 1: The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical M present and inversely proportional to the concentration of chemical N present. If the concentration of chemical N is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical M required to keep the reaction rate unchanged?

(A) 100% decrease

(B) 50% decrease

(C) 40% decrease

(D) 40% increase

(E) 50% increase

Solution:

Rate/M^2 = k

Rate*N = k

Rate*N/M^2 = k

If Rate has to remain constant, N/M^2 must remain the same too.

If N is doubled, M^2 must be doubled too i.e. M must become ?2 times. Since ?2 = 1.4 (approximately),

M must increase by 40%.

Answer (D)

Simple enough?

Very good explanation on the topic Karishma!

Thanks!

Karishma, in this problem:

In a certain business, production index p is directly proportional to efficiency index e, which is in turn directly proportional to investment i. What is p if i = 70?

(1) e = 0.5 whenever i = 60

(2) p = 2.0 whenever i = 50

Source: http://gmatclub.com/forum/in-a-certain-business-production-index-p-is-directly-63570.html

Your explanation was:

“production index p is directly proportional to efficiency index e,

implies p = ke (k is the constant of proportionality)

e is in turn directly proportional to investment i

implies e = mi (m is the constant of proportionality. Note here that I haven’t taken the constant of proportionality as k here since the constant above and this constant could be different)

Then, p = kmi (km is the constant of proportionality here. It doesn’t matter that we depict it using two variables. It is still just a number)”

How does this problem differ from the problems in this blog?

I followed the blog solution and arrived at

p/e = k and e/i = k

hence, pi/e = k.

Please advice,

Thanks.

This question is discussed here : http://www.veritasprep.com/blog/2015/06/important-caveat-on-joint-variation-gmat-questions/

I have also discussed why we use different methods in the two cases.

4. x varies directly with y^2 and y varies directly with z.

x/y^2 = k

y/z = k which implies that y^2/z^2 = k

Joint variation: x*z^2/y^2 = k

Shouldn’t it be

x/(y^2 * z^2) = k??

Because if x is directly proportional to y^2 and y^2 is directly proportional to z^2.

This implies x is directly proportional to z^2 and combining them, x is directly proportional to y^2 and z^2.

Check this post: http://www.veritasprep.com/blog/2015/06/important-caveat-on-joint-variation-gmat-questions/

It will answer your question.