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

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?

(1) e = 0.5 whenever i = 60
(2) p = 2.0 whenever i = 50

Bunuel · Accepted Answer

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.

KarishmaB · Answer

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).

prasannar · Answer

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

gettinit · Answer

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.

russ9 · Answer

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

Bunuel · Answer

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.

francoimps · Answer

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

KarishmaB · Answer

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.

francoimps · Answer

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?

KarishmaB · Answer

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 k eep the reaction rate unchanged ?

You need relation between N and M when reaction rate is constant.

harishbiyani8888 · Answer

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.

OptimusPrepJanielle · Answer

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

KarishmaB · Answer

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.

akbankit · Answer

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!!

Bunuel · Answer

Kinshook · Answer

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

TheUltimateWinner · Answer

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__

KarishmaB · Answer

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.

ScottTargetTestPrep · Answer

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

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

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

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