In a certain industry, production index x is directly propor

Question

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

(A) 100% increase

(B) 50% increase

(C) 30% increase

(D) 30% decrease

(E) 50% decrease

Source: Gmat Hacks 1800

KarishmaB · Accepted Answer

I have discussed Variation in this video in detail: https://youtu.be/AT86tjxJ-f0 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)

Marcab · Answer

The entire expression can be written in the form  x=k*y^2/z . When z is halved, the expression becomes  x=2k*y^2/z . In order to make this expression equal to its initial form, we need to divide this expression by 2. This can be done by  reducing   y  to  y/\sqrt{2} .
Therefore percent change=  y-y/1.414 => 0.414y/1.414 . This is almost equal to 30%.
Hope this helps.

Bunuel · Answer

Original question from GMAT Prep: Discussed here: the-rate-of-a-certain-chemical-reaction-is-directly-90119.html (I guess many Gmat Hacks questions are just reworded problems from GMAT Prep).

Carcass · Answer

you are correct ........

ziko · Answer

Normally i plug numbers to see what happens if certain change applies, i am not sure whether i am doing correct but please comment if anything wrong in my method:

Let say the efficiency index y=10 and investment index=1, so, per question terms, production index  x=y^2/x=10^2/1=100 ; now we change the investment index to half (i assumed that it has been decreased to half since it is not clear in the question) and we have the following:  100=y^2/0.5=(y^2)*2=100  -->  y=\sqrt{50}  --> equals to about  7 , then we can see that efficiency index should be reduced to approximately  30%  (from 10 to 7).

I hope that my way of thinking is correct.

Transcendentalist · Answer

I used the same method but skipped the calculations since you know that since z is being halved y also has to decrease. There are only 2 options that decrease. If i plug in 50, the condition isn't met and hence 30 is the answer. You should be able to deduce this in less than 30 secs.

ScottTargetTestPrep · Answer

“Production index x is directly proportional to the square of efficiency index y, and indirectly proportional to investment index z” means x = k * y^2/z for some positive value k.

Now we are given that z becomes z/2 and we can let y become ty for some positive value t. So we have x = k * (ty)^2/(z/2). Since we are keeping x the same and recall that x = k * y^2/z, so we have:

k * y^2/z = k * (ty)^2/(z/2)

y^2/z = t^2 * y^2 * 2/z

1 = t^2 * 2

1/2 = t^2

t = √(1/2) ≈ 0.7

Thus we see that the efficiency index y has to become 0.7y, or a 30% decrease, in order to keep the production index x the same.

Answer: D

flowertown · Answer

how to easily know that 't = √(1/2) ≈ 0.7' please?

D4kshGargas · Answer

Are we really supposed to perform such calculation?
I mean  \frac{1}{\sqrt{2}}  ?

hassan233 · Answer

How I did it:
Since X is is directly proportional to y^2 and inversely proportional to z, we can say X = (y^2)/z
If we are cutting z in half, then we have X = (y^2)/((1/2)*z).
So we are multiplying the bottom by 1/2, this means for X to stay the same, we need to multiply y by y^2 by half.
So now we have ((1/2)*y^2)/((1/2)*z)

Let us try to combine 1/2 and y^2 together (from (1/2)*y^2))

So we need to somehow make 1/2 into a number (let's say h) squared. So we need h^2 = 1/2.
Let's square root both sides, now we have h = sqrt(1/2) = sqrt(1)/sqrt(2) = 1/1.4
YOU SHOULD HAVE sqrt(2) AND sqrt(3) memorized. 
Valentines day is February 14th, 2/14 => sqrt(2) = 1.4
St Patricks Days is March 17th, 3/17 => sqrt(3) = 1.7

Anyways,
h = 1/1.4, so now we have: (((1/1.4)^2)*(y^2))/((1/2)*z) = (((1/1.4)*y)^2)/((1/2)*z).
So we are reducing y to (1/1.4) of y
So we know it is a decrease. By how much?
Well, 1 - (1/1.4) = 1 - 5/7 = 2/7

You know 2/7 is NOT 50% (3.5/7 is)

The only other decrease is 30%, if 3.5/7 is 50%, then you can eyeball that 2/7 is 30%ish.

Hope that helped.

Kinshook · Answer

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?

Production Index x = k y^2/z ; y is efficiency index; z is investment index

Since production index x is same 
  x = \frac{ky_1^2}{z_1} = \frac{ky_2^2}{z_2}

\frac{z_2}{z_1} = \frac{1}{2}

x = \frac{k}{2} = \frac{ky_2^2}{y_1^2}  
 \frac{y_2^2}{y_1^2} = \frac{1}{2} 
 y_2/y_1 = \sqrt{.5} = .70 = 1- .30

IMO D

In a certain industry, production index x is directly propor

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