In a certain business, production index p is directly proportional to

\(a\) is directly proportional to \(b\) means that as the absolute value of \(b\) gets bigger, the absolute value of \(a\) gets bigger too, so there is some non-zero constant \(x\) such that \(a=xb\);

So if \(a\) is directly proportional to \(b\) (\(a=xb\)), then vise-versa is also correct: \(b\) is directly proportional to \(a\) (\(b=\frac{1}{x}*a\) as the absolute value of \(a\) gets bigger, the absolute value of \(b\) gets bigger too).

\(a\) is inversely proportional to \(b\) means that as the absolute value of \(b\) gets bigger, the absolute value of \(a\) gets smaller, so there is some non-zero constant constant \(y\) such that \(a=\frac{y}{b}\).

So if \(a\) is inversely proportional to \(b\) (\(a=\frac{y}{b}\)), then vise-versa is also correct: \(b\) is inversely proportional to \(a\) (\(b=\frac{y}{a}\) as the absolute value of \(a\) gets bigger, the absolute value of \(b\) gets smaller).

As for the question:
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?

Given: \(p=ex\) and \(e=iy\) (for some constants \(x\) and \(y\)), so \(p=ixy\). Question: \(p=70xy=?\) So, basically we should find the value of \(xy\).

(1) e = 0.5 whenever i = 60 --> as \(e=iy\) then \(0.5=60y\) --> we can find the value of \(y\), but still not sufficient.
(2) p = 2.0 whenever i = 50 --> as \(p=ixy\) then \(2=50xy\) --> we can find the value of \(xy\). Sufficient.

Answer: B.

Hope it's clear.
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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)

e.g. if p = 2e and e = 3i
p = 6i will be the relation. 6 being the constant of proportionality.

So if you have i and need p, you either need this constant directly (as you can find from statement 2) or you need both k and m (statement 1 only gives you m).
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we need P when i is some value...

we know p is dependent on e and e is dependent on i

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 -> does not give the value or relation between e and P thus insufficient
2) p = 2.0 whenever i = 50 -> gives the relation between p and i thus we can find p when i=70

thus B

Would p be directly proportional to i as well if e is proportional to p? I am thinking it should be, however the constant proportion will be different between p and e and e and i and thus entirely separate between p and i? thanks.

Hi Bunuel,

When you break it down like that, it makes complete sense but I made the following error. Can you please clarify why this isn't true?

\(\frac{p}{e}\) = \(\frac{e}{i}\)

\(\frac{p}{.5}\) = \(\frac{.5}{60}\) and solve for p. If the ratios are proportional, shouldn't .5/60 give me a relationship for p/e since I already know E? This led me to choose "D" as the answer choice.

Thanks

Directly proportional means that as one amount increases, another amount increases at the same rate.

We are told that p is directly proportional to e and e is directly proportional to i. But it does NOT mean that the rate of increase, constant of proportionality, (x in my solution) for p and e is the same as the rate of increase, constant of proportionality, (y in my solution) for e and i.

Or simply put, we have that \(\frac{p}{e}=x\) and \(\frac{e}{i}=y\) but we cannot say whether x=y, so we cannot say whether \(\frac{p}{e}=\frac{e}{i}\).

Hope it's clear.
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Hi Karishma,

If I were to follow the solution for your post on joint variations in this blog https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html/2013/02 ... mment-5837,

I would arrive with the solution:
p/e = k and e/i = k
hence, pi/e = k is the joint variation.

Why does this problem differ?

Thanks

Joint variation gives you the relation between 2 quantities keeping the third (or more) constant.
p will vary inversely with i if and only if e is kept constant.

Think of it this way, if p increases, e increases. But we need to keep e constant, we will have to decrease i to decrease e back to original value. So an increase in p leads to a decrease in i to keep e constant.
But if we don't have to keep e constant, an increase in p will lead to an increase in e which will increase i.

Here, we are not given that e needs to be kept constant. So we will not use the joint variation approach.
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Hi Karishma,

Thanks for your reply.

How will I know whether the question asks that a certain variable needs to be kept constant?

The question above, "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?" seems similar to the question on your blog post, "x varies directly with y and y varies inversely with z."

What should I explicitly look for to determine whether the issue is joint variation or not?

It will be told that the third variable has to be kept constant.

Note how the independent question is framed in my post:
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?

You need relation between N and M when reaction rate is constant.
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Hi Krishna/Bunuel,

Can you explain why the constant k could be different? I took constant k for both the proportionalities and marked answer D.

Given: p = c1*e - (i) where c1 is a constant
and e = c2*i - (ii) where c2 is a constant

Required: p = ? when i = 70
p = c1*e = c1*c2*i
p = c1*c2*70 - (iii)
Hence we need to find the value of c1 and c2

Statement 1: e=0.5 whenever i=60
From (ii),
0.5 = c2*60
Clearly we cannot solve for c1
INSUFFICIENT

Statement 2: p=2.0 whenever i=50
From (i),
2 = c1*50
and we know that
p = c1*70

Dividing both the equations,
2/p = 50/70
Hence we can solve for p
SUFFICIENT

Correct Option: B

Let me ask you the flip question: why do you think both constants would have the same value?

"In a certain business, production index p is directly proportional to efficiency index e," - say, p = 2e so when e doubles, p becomes four times etc

"e is in turn directly proportional to investment i." - Now how does this imply that e = 2i? We could very well have e = 3i or e = i/2 etc

We are not given that whatever the relation is between p and e, the relation has to be the same between e and i too.
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Hi Bunuel/Karishma

I solved this question and culminated in choice D as answer choice. I am posting my solution here, please help me with reason why i am wrong.

Statement 1 - e= 0.5 whenever i =60

per question stem we know that p is directly proportion to e and e is directly proportion to i , using this we can calculate P as 60*0.5 = 30 at e and i being 0.5 and 60 respectively; and to find the value of e (called e1) at i =70 . e = (0.5/60) *70 ---> 7/30 is the value of e (e1) at i = 70.

Now we can calculate p at i =70 using above relation = 30* (7/6) * (7/15)

where 7/6 is constant for i and 7/15 {7/(30*0.5)} is constant for e.

Using above i arrived at option D which is wrong. Please let me know me the pitfalls and where did i move away from the relevant concept.

Looking forward to hear from the Masters!!

So, you are calculating p as p = ei, which is wrong. \(p=ex\) and \(e=iy\) (for some constants \(x\) and \(y\)), so \(p=ixy\).

Not sure what you are doing in the part in red.

Please check correct approaches above. They should help to get the flaws in your logic.

Theory:
Variations On The GMAT - All In One Topic

Questions:
https://gmatclub.com/forum/a-is-directly ... 88971.html
https://gmatclub.com/forum/the-rate-of-a ... 90119.html
https://gmatclub.com/forum/if-the-price- ... 50508.html
https://gmatclub.com/forum/in-a-certain- ... 46815.html
https://gmatclub.com/forum/a-spirit-and- ... 68909.html
https://gmatclub.com/forum/in-a-certain ... 80941.html
https://gmatclub.com/forum/a-certain-qu ... 47469.html
https://gmatclub.com/forum/recently-fue ... 44188.html
https://gmatclub.com/forum/the-cost-of- ... 44190.html
https://gmatclub.com/forum/the-price-of ... 44191.html
https://gmatclub.com/forum/if-the-ratio ... 44184.html
https://gmatclub.com/forum/20-workmen-c ... 44185.html
https://gmatclub.com/forum/the-variable ... 05761.html
https://gmatclub.com/forum/in-a-certain ... 63570.html
https://gmatclub.com/forum/the-amount-of ... 93667.html

Check below for more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
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Given: In a certain business, production index p is directly proportional to efficiency index e, which is in turn directly proportional to investment i.

Asked: What is p if i = 70?

p = k1 * e
e = k2 * i
p = k1 * k2 * i = k * i where k = k1 * k2

p = 70 k = 70 k1 * k2 if i = 70

(1) e = 0.5 whenever i = 60
e = k2 * i
.5 = k2 * 60
k2 = .5/60 = 1/120
Since k1 is unknown
NOT SUFFICIENT

(2) p = 2.0 whenever i = 50
p = k* i
2 = k * 50
k = 2/ 50 = 1/25
p = 70 k = 70 /25 = 2.8
SUFFICIENT

IMO B
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Hello VeritasKarishma
Thanks for the explanation with kudos.
-->Which one we have to do with i? Should we decrease i or fixed the previous value of i?
Like,
if p=10; e=7; i=15
If we increase the value of those by 2 we get-->
p=12; e=9; i=17
If we decrease i by 2 to decrease e, we get-->
p=12; e=7 (original value); i=15 (original value)
Are you talking about the blue part?
Thanks__
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Asad, this is not a joint variation question and we don't have to use it.

This point is discussed here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html/2015/0 ... questions/

What happens in Joint Variation?

In joint variation, if p is directly proportional to e and e is directly proportional to i, then p is inversely proportional to i.
Note why: This relation of p and i holds when e is constant.
If we increase p, e increases (because e is directly proportional to p).
So what happens to i? Relation between p and i depends on keeping e constant. Since e has increased, it needs to be reduced back to keep it constant. So we should reduce i to reduce e (because i is directly proportional to e).

As per joint variation, pi/e = constant
If p doubles, i should become (1/2) to maintain the value of pi/e.

This question does not say that e needs to be held constant. So the impact will be in sequence. If p doubles, e doubles. If e doubles, i doubles. So p is directly proportional to i.
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Solution:

We need to determine the value of p when i = 70. We are told that p is directly proportional to e, which is in turn directly proportional to i. Recall that if x is directly proportional to y, then x = ky for some positive constant k.

Therefore, we have p = ke and e = ji for some positive constants k and j. In other words, p = kji, and if we can determine the values of k and j, or the value of kj, then we can determine the value of p.

Statement One Alone:

Since e = ji, we have:

0.5 = j(60)

j = 0.5/60 = 1/120

However, since we still don’t know the value of k, we can’t determine the value of p. Statement one alone is not sufficient.

Statement Two Alone:

Since p = kji, we have:

2.0 = kj(50)

kj = 2/50 = 1/25

Since kj = 1/25 and p = kji, then, if j = 70, we see that p = 1/25 * 70 = 14/5 = 2.8. Statement two alone is sufficient.


Answer: B

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