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Manager  Joined: 28 May 2009
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In a certain industry, production index x is directly propor  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 49% (02:26) correct 51% (02:28) wrong based on 324 sessions

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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

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Re: In a certain industry, production index x is directly  [#permalink]

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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%)

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Re: In a certain industry, production index x is directly  [#permalink]

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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

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.
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Re: In a certain industry, production index x is directly propor  [#permalink]

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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

Original question from GMAT Prep:

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

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase

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).
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Re: In a certain industry, production index x is directly propor  [#permalink]

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Bunuel wrote:
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

Original question from GMAT Prep:

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

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase

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).

you are correct ........
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Re: In a certain industry, production index x is directly propor  [#permalink]

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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.
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Re: In a certain industry, production index x is directly  [#permalink]

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Marcab wrote:
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.

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.
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Re: In a certain industry, production index x is directly propor  [#permalink]

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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

“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.

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Re: In a certain industry, production index x is directly propor  [#permalink]

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ScottTargetTestPrep wrote:
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

“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.

how to easily know that 't = √(1/2) ≈ 0.7' please? Re: In a certain industry, production index x is directly propor   [#permalink] 12 Sep 2018, 08:46
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