In a certain industry, production index x is directly propor

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).

you are correct ........

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.

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.

“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

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

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

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.

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