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To rent an office, each member of a club must pay n dollars. If two mo

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New post 28 Nov 2017, 22:31
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To rent an office, each member of a club must pay n dollars. If two more members join the club, the per-member payment would be reduced by two dollars. Which of the following could be the number of members currently in the club?

I. 16
II. 17
III. 18

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
[Reveal] Spoiler: OA

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Kudos [?]: 135673 [0], given: 12706

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Kudos [?]: 403 [1], given: 645

To rent an office, each member of a club must pay n dollars. If two mo [#permalink]

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New post 29 Nov 2017, 10:34
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Bunuel wrote:
To rent an office, each member of a club must pay n dollars. If two more members join the club, the per-member payment would be reduced by two dollars. Which of the following could be the number of members currently in the club?

I. 16
II. 17
III. 18

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

I went off sheer intuition here and tested numbers for Option I, then saw a pattern. The answer I got: all three are possible. Please correct me if I am mistaken.

I. 16

If there are 16 members, add 2 (=18), and multiply 16 * 18.

Do not find LCM; we need the product of 16 and 18 so that when we divide the product (total $) by (16 + 2), we will get $16.

16 members * $18 ea = $288
18 members would pay $288/18 = $16 each

16 members pay $18 each
18 members pay $16 each
2 more members, $2 less per person

II. 17
If there are 17 members, add 2 and multiply 17 * $19 = $323
19 members will pay $323/19 = $17 each

17 members pay $19 each
19 members pay $17 each
2 more members, $2 less per person

III. 18
Add 2. Multiply 18 * $20 ea = $360
20 members would pay $360/20 = $18 each

18 members pay $20 each
20 members pay $18 each
2 more members, $2 less per person

All three are possible.

Answer E

*I started algebraically, but its implications befuddled me, so I switched to testing numbers.

A * n = S, where S must remain the same
A = amount paid per person
n = number of members
x = $ per person (i.e., is A)

Original:
x * n = xn = SUM

New
(x - 2)(n + 2) = same SUM

xn = (x - 2)( n + 2)
xn = xn + 2x - 2n - 4
4 = 2x - 2n
4 = 2(x - n)
2 = x - n
n + 2 = x

Don't laugh too hard if I'm over the cliff. I think this means the same as what I did instinctively: add 2 to the original number of members to get a dollar amount for what each pays originally.

That dollar amount turns out to be the new number of total members.

Kudos [?]: 403 [1], given: 645

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Re: To rent an office, each member of a club must pay n dollars. If two mo [#permalink]

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New post 01 Dec 2017, 06:53
Bunuel wrote:
To rent an office, each member of a club must pay n dollars. If two more members join the club, the per-member payment would be reduced by two dollars. Which of the following could be the number of members currently in the club?

I. 16
II. 17
III. 18

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III


If we let m = the number of members in the club, then the total money they will pay for the rental is mn. When two more members join, the number of members is now (m + 2), and now each person pays two dollars less (n - 2). Thus,we can create the equation:

mn = (m + 2)(n - 2)

mn = mn + 2n - 2m - 4

2m = 2n - 4

m = n - 2

We see that the number of members in the club is 2 fewer than the number of dollars contributed per person. Since there is no restriction on the value of n (as long as n is greater than 2), we see that m can be any of the three given values in the Roman numerals.

Answer: E
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Kudos [?]: 1019 [0], given: 3

Re: To rent an office, each member of a club must pay n dollars. If two mo   [#permalink] 01 Dec 2017, 06:53
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