A farmer wants to allocate her expenditures to maximize her profits

Question

A farmer wants to allocate her expenditures to maximize her profits during the coming year. Let  C  be her spending on cropland,  F  her spending on fertilizer,  L  her spending on labor,  M  her spending on machinery, and  S  her spending on seed. She has estimated that, if she allocates at least 100 euros to each of these categories, then her profit during the coming year will be ( C  – 97)( L  – 92)( S  – 95)( M  – 87)( F  – 90) euros. She wishes to calculate the benefits of expenditure greater than 100 euros in one these categories, if she were to spend exactly 100 euros in each of the others.

Select for  Greatest returns  the resource for which, according to the desired calculation from the model, each euro of spending beyond 100 euros is predicted to make the greatest contribution to profits (or least contribution to losses). And select for  Least returns  the resource for which, according to the desired calculation, each euro of spending beyond the minimum necessary amount is predicted to make the least contribution to profits (or greatest contribution to losses). Make only two selections, one in each column.

ID: 700192

KarishmaB · Accepted Answer

Profit = ( C  – 97)( L  – 92)( S  – 95)( M  – 87)( F  – 90) euros
Assume 100 euros spent on each category.

Profit = 3 * 8 * 5 * 13 * 10

Now think: She wants to allocate 1 euro extra. Where should she allocate it? Should she give it to C to get 4 as the first term or to say M to get 14 as the second last term? By allocating 1 euro to 3, she is increasing it by 33.3% i.e.

Profit = (3 * 133/100) * 8 * 5 * 13 * 10
So essentially, the profit has gotten multiplied by 133/100.

On the other hand, if she were to allocate 1 euro to 13, then she would increase it by about 7% to 8%

Profit = 3 * 8 * 5 * 13 * (107/100)* 10
So essentially, the profit has gotten multiplied by 107/100.

Hence, to  MAXIMIZE profit , she should allocate additional money to C Cropland and to  MINIMIZE  she should allocate it to M Machinery. 
ANSWER.

Method 2 : Think about it this way:

Say I want to double the profit: 
Profit = 3 * 8 * 5 * 13 * 10  * 2

How much money do I need to do this?
If I double 3 (Profit = 6 * 8 * 5 * 13 * 10), I need just 3 additional euros
but if I double 13 (Profit = 3 * 8 * 5 * 26 * 10), I need 13 additional euros.

So 3 will give maximum increase in profit with minimum money and 13 will give minimum increase in profit.

Here is  a video discussing how to handle Two Part Analysis Questions .

egmat · Answer

Since you are reading this post, its fair to assume that you have solved this question. Before I give you the video solution, I want you to tell me your experience with this question:
 
 Did you read this question and got scared of the profit expression? 
 Did you brave through the profit expression but reached the question stem and then gave up on this question? 
 Or did you brave through the complex language and processed the question but ended up selecting the wrong answer?

Now watch the solution and observe the following:
 
 This question is the testament of the translate process skill – do not give up when you see dense language. Instead read that with the mindset of ‘owning the dataset’. 
 This question exemplifies of drawing proper inferences and arriving at proper approach so that you can arrive at the correct answer.

https://www.youtube.com/watch?v=wH73fbQjPD4

Would love to hear your thoughts on it.

chetan2u · Answer

She has estimated that, if she allocates at least 100 euros to each of these categories, then her profit during the coming year will be (C – 97)(L – 92)(S – 95)(M – 87)(F – 90) euros.

Further all except  one  are 100, and this  one  is >100.

Let all be 100 => (C – 97)(L – 92)(S – 95)(M – 87)(F – 90)
(100 – 97)(100 – 92)(100– 95)(100 – 87)(100 – 90)
3 * 8 * 5 * 13 * 10

We have to make the above highest by increasing one of them, the one that is >100.
Let us check with extremes
3*(8*5*13*10) = 3*5200….if increased by 1, then 4*5200 = 20800
(3*8*5*10)*13= 1200*13….if increased by 1, then 14*1200 = 16800

So increasing C makes it the largest and increasing M makes it smallest.

Let us check logically too  
Let us see what happens when we add some positive number x to each one by one.
A. 3 * 8 * 5 * 13 * 10 becomes (3+a) * 8 * 5 * 13 * 10 =  3 * 8 * 5 * 13 * 10  + a * 8 * 5 * 13 * 10
B. 3 * 8 * 5 * 13 * 10 becomes 3 * (a+8) * 5 * 13 * 10 =  3 * 8 * 5 * 13 * 10  + a * 3 * 5 * 13 * 10
C. 3 * 8 * 5 * 13 * 10 becomes 3 * 8 * (a+5) * 13 * 10 =  3 * 8 * 5 * 13 * 10  + a * 3 * 8 * 13 * 10
D. 3 * 8 * 5 * 13 * 10 becomes 3 * 8 * 5 * (a+13) * 10 =  3 * 8 * 5 * 13 * 10  + a * 3 * 8 * 5 * 10
E. 3 * 8 * 5 * 13 * 10 becomes 3 * 8 * 5 * 13 * (a+10) =  3 * 8 * 5 * 13 * 10  + a * 3 * 8 * 5 * 13
 The colored portion is same for all, so concentrate on only the second part. In the second part 'a' replaces the number to which it is added. So, when a replaces smallest number 3, the product will be the maximum, and when a replaces the largest number 13, the product will be the least

Elite097 · Answer

KarishmaB  i got the reverse and cant visualise why we are getting reverse solution . Increasing a higher number would ideally give a larger product

chetan2u · Answer

I am sure  KarishmaB  too would pitch in. But how did you get the conclusion.

Say, you have a product 3*5 or 15
Increase lower number: 4*5 = 20
Increase the higher number: 3*6 = 18
Which is greater??

Arithmetically: When you compare the two
(3+1)*5 and 3*(5+1) 
3*5+1*5 and 3*5+1*3
3*5+1*5 > 3*5+1*3

mayankdgmat · Answer

For easy to understand let us take three numbers 1,2,3 and each need to be raised by 1 keeping other two constant. Need to find max and min

(1+1)*2*3 = 1*2*3+1*2*3 so increase by 6

similarly 1*(2+1)*3 increase by 3

1*2*(3+1) increase by 2

so increasing least number is increasing profit by higher return and vice versa

Anonymous · Answer

It didn't show the image before, let me know if it's ok now.

ShubhangiGoel · Answer

chetan2u · Answer

You should take an average of 100-120 seconds, that is close to 2 minutes or slightly less.
Try to finish the easy ones in a minute so that to have more time for review and tougher ones.

Eli0025 · Answer

It's just taking the derivatives; whichever case gives you the highest derivative is the greatest rate of return. In this case, Cropland is the greatest because the derivative is highest if you differentiate with respect to C and force the other four variables to 100. Same case with Machinery.

UDEMYtutorJackson · Answer

SHORTCUT METHOD

Given Profit = (C – 97)(L – 92)(S – 95)(M – 87)(F – 90)

Spend a few seconds making observations. Notice that among the 5 terms (C – 97),(L – 92),(S – 95),(M – 87),(F – 90) the only thing that can differentiate one from another is the amount subtracted .  There is no other difference  ( such as the coefficient or so).

Hence the answer has to be between (C – 97) and (M – 87) or in other words between C and M (because from C the max amount is subtracted and from M the min amount is subtracted. We are just looking at the max and min subtraction , which is the only differentiating factor). We, as of now, do not know which one will have the greatest impact and which one has the lowest impact.

To find out, just substitute and check the 2 possibilities.

Possibility 1: C = 100 and let's take M =200 ( 200 or any value greater than 100. Just taking 200 to make the calculation easy)
Profit = 3 * 113
Possibility 2 : C =200 and M = 100
Profit = 103 * 13

Clearly Possibility 2 gives greater Profit.

Hence allocating more to C will give the greatest return. And as you can see form Possibility 1, allocating more to M will give the Least return.

PS: Do Check out my GMAT courses on Udemy for more tricks.

Oppenheimer1945 · Answer

Trick to not fall for swap answer:

Take F(x,y)=(x-97)(y-87).

Find F(101,100)=52 and F(100,101)=42

You get higher return when you tweek the term with higher constant term,

Hence C for maxima, and M for min

nisen20 · Answer

don't be fooled by those 5 horrible variables. make a simpler example for yourself.

this farmer has two tasks burning money: task A costs 6 and task B cost 8.
originally, she plans to spend 10 for each task, and the profits is 4 × 2 = 8

now, she wants to spend more on one of two tasks to make more money. let's make it 15.
if she spend 15 on task A, the profits will be 9 × 2 = 18
if she spend 15 on task B, the profits will be 4 × 7 = 28

see? always spend the extra money on the more expensive task.

Aalto700 · Answer

How can this question belogn to 800+ level  chetan2u ?

Profit = (C – 97)(L – 92)(S – 95)(M – 87)(F – 90) euros
Assume 100 euros spent on each category.
Profit at 100eur = 3 * 8 * 5 * 13 * 10 = A, which is constant.

Now change one factor by x+k, new profit is A / x * (x+k) = A * (1 + k/x)

Find x for the min and max of 1 + k/x, which k is constant.

KarishmaB  : 
some GMAT teachers in my country said that you often have high-level answers.

chetan2u · Answer

The difficulty level gets adjusted automatically depending on the timer used. This particular question has almost 50% wrong answers.

KarishmaB · Answer

Ah! Thank you! I am happy to know that teachers in your country like my solutions.

PeachSnapple1 · Answer

Fairly easy question, took me sub one minute.

Let's say that a genie gave me a wish to multiply one bill of my choice numerous times, what bill would I choose? Of course I would choose Mr. Ben Franklin ($100), and not any other bills ($50, $20, $10, etc)
The same thing applies here. Let's say I got a product: P = A*B (where one is a constant - I will call it "the base", the other can change value). If I want to max P, I will max the base and NOT the variable. 
Now, in the prompt we have a product: P = (C – 97)(L – 92)(S – 95)(M – 87)(F – 90)
If I want to maximize P, then I will choose the maximum base and leave out the one with a smallest value. Assume I spent exactly $100 on each category, then the smallest value would be  (C-97), which is also the answer for maximum return . Applying the same thing and I will find  (M-87) giving me minimum return.

A farmer wants to allocate her expenditures to maximize her profits

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Greatest returns	Least returns
		Cropland
		Fertilizer
		Labor
		Machinery
		Seed

GMAT Two-part Analysis (TPA) Questions

A farmer wants to allocate her expenditures to maximize her profits

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