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

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Math Expert
Joined: 02 Sep 2009
Posts: 50077
To rent an office, each member of a club must pay n dollars. If two mo  [#permalink]

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07 Feb 2016, 10:40
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45% (medium)

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62% (02:29) correct 38% (02:26) wrong based on 45 sessions

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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

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Joined: 02 Aug 2009
Posts: 6985
Re: To rent an office, each member of a club must pay n dollars. If two mo  [#permalink]

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07 Feb 2016, 22:59
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

Hi,

info from Q

There are no retrictions on either number of members or amount of payment..
Increase of 2 members results in contribution reducing by 2 dollars..

As member and contribution can be changed, all integers should satisfy the member number..
so E..

BUT lets see by ALGEBRIC way..

let the number of member =x and the contribution be n per member..
so total cost= xn..

in the secon scenario, x increases by 2 and n decreases by 2..
so cost= (x+2)(n-2)..

cost is he same both times, so:-
xn=(x+2)(n-2)..
xn=xn+2n-2x-4..
or 2x=2n-4..
x=n-2...

hence x can take any values, as n changes..
E
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Math Expert
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To rent an office, each member of a club must pay n dollars. If two mo  [#permalink]

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28 Nov 2017, 23:31
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To rent an office, each member of a club must pay n dollars. If two mo  [#permalink]

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29 Nov 2017, 11:34
1
1
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.

*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.

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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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01 Dec 2017, 07:53
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.

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Re: To rent an office, each member of a club must pay n dollars. If two mo &nbs [#permalink] 01 Dec 2017, 07:53
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