megafan wrote:
In a certain industry, production index x is directly proportional to the square of efficiency index y, and indirectly proportional to investment index z. If a business in this industry halves its investment index, which of the following is closest to the percent change in the business’s efficiency index required to keep the production index the same?
(A) 100% increase
(B) 50% increase
(C) 30% increase
(D) 30% decrease
(E) 50% decrease
Source: Gmat Hacks 1800
I am discussing direct and inverse variation on my blog nowadays.
http://www.veritasprep.com/blog/2013/01 ... -directly/http://www.veritasprep.com/blog/2013/02 ... inversely/Next post will discuss a similar question using both. I have tried to show every step here to avoid confusion but on the blog, I am going to be more direct.
By the way, the question is not very GMAT-like.
\(x = ky^2\) (x is directly proportional to y^2. k is any constant)
\(xz = k\) (I am assuming that indirectly is actually inversely proportional)
Both together, \(\frac{xz}{y^2} = k\) (Assume z to be constant, you get x is directly proportional to y^2. Assume y to be constant, you get x is inversely proportional to z)
\(\frac{x_1*z_1}{(y_1)^2} = \frac{x_2*z_2}{(y_2)^2}\)
Given that \(z_2 = (\frac{1}{2})z_1\) and that \(x_1\)should be equal to \(x_2\). What is the relation between \(y_1\) and \(y_2\)?
\(\frac{x_1*z_1}{(y_1)^2} = \frac{x_1*(1/2)z_1}{(y_2)^2}\)
\((y_2)^2 = \frac{(y_1)^2}{2}\)
\(y_2 = \frac{y_1}{\sqrt{2}}\)
\(y_2 = \frac{y_1*\sqrt{2}}{2}\)
\(y_2 = 0.7*y_1\)
(A decrease of 30%)
Answer (D)