A certain company has 500 employees split among three divisions, such

Question

A certain company has 500 employees split among three divisions, such that there are at least 100 employees in each division. An $8,000,000 benefits budget is to be distributed among the three divisions, and a division will be considered underfunded if it receives no more than $10,000 per employee, and overfunded if it receives at least $40,000 per employee. How many divisions of the company are overfunded?

(1) If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is underfunded.

(2) If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is overfunded.

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arkaja11 · Answer

Looks scary to be honest, but probably it's a tricky question. In my opinion, its option D (Both are alone sufficient)

Let us first put all the parameters:

The total number of employees is 500, split among three divisions (A,B,C) with each division having at least 100 employees. The total benefits budget is $8,000,000.

We denote the number of employees in division A,B and C as a,b and c with a+b+c = 500 and a,b,c ≥100

We denote the budget to divisions in division A,B and C as x,y and z with x+y+z = $ 8000k

We need to check how many of these divisions receive at least $40,000 per employee:

How many of (x/a , y/b , z/c ) ≥ $ 40k

Lets look at Statement 1 :

"If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is underfunded"

This implies : x/a > $ 10k and y/b > $ 10k and z/c > $ 10k

Supposing one division to be just overfunded -
Let's say division A has least number of employees = 100 and receives benefits of $ 40k/employee. Hence, x = $ 4000k

This will imply --> y+z = $ 4000k and b+c = 400 (from above constraints)

For any combination, it's not possible for both B and C to not stay underfunded.

(Let's take y/b = $ 11k and b = 200, then c = 200 and z/c = $ 9k (underfunded))

Hence, if 1 division is overfunded, atleast 1 division will be underfunded.

So, we can conclude that for any arrangement where "there is no possible rearrangement in which any division is underfunded", number of overfunded divisions = 0

Hence statement 1 is sufficient.

Lets look at Statement 2 :

"If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is overfunded."

x/a < $ 40k, y/b < $ 40k or z/c < $ 40k

Since this ensures that no division is overfunded in any rearrangement, it implies no division is overfunded with the original budget distribution.

Hence statement 2 is also sufficient.

We can conclude that both the statements are sufficient alone. Hence Option D.

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tgsankar10 · Answer

Let amount distributed to each division be  x, y, z  and no of employees in each division be  a, b, c

x+y+z=4000k  and  a+b+c=500

Min no of employees = 100, no maximum provided

Statement 1:  "If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is underfunded"

Let's assume that one division is overfunded in original arrangement. With min no of employees 100

x/a ≥ 40k  i.e  x/100 ≥40k . So x will be  4000k

This brings  y+z=4000k  and  b+c=400

With this condition, another division can not be over funded i.e 4000k/100 where 3rd division will be left without any funds. At this point no of divisions overcrowded  ≤1

Let's assume, 2nd division is underfunded. Now the distribution will become:

4000k/100 + 1000k/100 + 3000k/200

Rearranging this in another two possibilities will leave a division in one of them. Even if we distribute the remaining funds and employees equally to 2nd & 3rd division we will end up in a similar scenario.

Possible rearrangements:

1000k/100 + 3000k/100 + 4000k/200 -  Fair distribution

3000k/100 + 4000k/100 + 1000k/200 -  Fair / overfunded / underfunded

Therefore, there can not be a division which is overfunded or no of divisions overfunded is  0

Hence, Statement 1 alone is sufficient.

Statement 2:  "If the amounts assigned to each division are rearranged such that each division receives an amount that had been assigned to another division, there is no possible rearrangement in which any division is overfunded."

Applying the same analysis as for statement 1 will give one overfunded possibility

So, in this case we should start with underfunded division, for which we don't have maximum no of employees per division.

Hence, Statement 2 alone is not sufficient.

Answer: A

DylanD · Answer

I've attempted max/min ratios of people (1 to 1 to 3 = 5) and of $ received (1 to 1 to 1 = 3 parts $ total), but I've spent far too long & only could guess that (2) would only suffice if there were an equal ratio of people/group.

shobhit770 · Answer

please advise the way to go about this question as i wasnt able to comprehend the problem

egmat · Answer

Hi shobhit770,

This question looks scary because of the "rearrangement" wording, but once you translate it into plain terms it's a clean sufficiency check. Let me give you a way in.

First, decode the setup. You have  3  divisions with employee counts a, b, c (each at least  100 , adding to  500 ) and three dollar amounts x, y, z (adding to  $8,000,000 ). "Per-employee funding" is just amount / employees. A division is  underfunded  if that ratio is  10,000  or less, and  overfunded  if it's  40,000  or more. The question only asks: how many divisions are overfunded?

Now the key to the wording. "Rearrange so each division receives an amount assigned to another division" means you swap the three dollar amounts onto different divisions. With exactly  3  divisions, the only such swaps are the two cycles (A->B->C->A and its reverse) - nobody can keep their own amount.

How to test each statement: use the worst-case pairing.

Statement (1): "no rearrangement leaves any division underfunded." Underfunding is most likely when the smallest amount lands on the largest division. If even that pairing stays above  10,000  per employee, then no division can be overfunded. Here's why: if some division were overfunded, it would hold at least  40,000  x  100  =  $4,000,000 , leaving at most  $4,000,000  for the other  400 + employees. You could then route a small amount onto a big division and force a ratio of  10,000  or below - i.e., a rearrangement  would  be underfunded. That contradicts the statement. So statement (1) forces the overfunded count to be  0 .  Sufficient .

Statement (2): "no rearrangement is overfunded." Here's the trap - this only constrains the two  swapped  arrangements, not the original. Because every division must give up its own amount, a division that is overfunded in the  original  setup is never tested by these swaps. So you can build a case with one division overfunded originally while no swap creates an overfunded division. The count could be  0  or it could be  1  - not pinned down.  Not sufficient .

The takeaway:  "rearrangement" here means a full swap (a derangement), so it never re-tests the original pairing. That single observation is what separates (1) from (2).

Answer: A

gullyboy09 · Answer

Hi  KarishmaB  can you help me with this question? What's the meaning of "each division receives an amount that had been assigned to another division"?

A certain company has 500 employees split among three divisions, such

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