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by vaibhav101 » Tue Sep 04, 2018 5:58 am

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a group of people met at a party. Each person shook hands with everyone else. Mr. Smith shook hands with 3 times as many men as women. Mrs. Smith shook hands with 4 times as many men as women. How many men and women were there at the party?

A 5 , 16
B 16 , 5
C 18 , 3
D 12 , 9
E 9 , 12
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by GMATGuruNY » Tue Sep 04, 2018 6:37 am
vaibhav101 wrote:a group of people met at a party. Each person shook hands with everyone else. Mr. Smith shook hands with 3 times as many men as women. Mrs. Smith shook hands with 4 times as many men as women. How many men and women were there at the party?

A 5 , 16
B 16 , 5
C 18 , 3
D 12 , 9
E 9 , 12
Mr. Smith shook hands with 3 times as many men as women.
The phrase in blue implies that the number of men who shook hands with Mr. Smith is a MULTIPLE OF 3.
Thus, the total number of men = (multiple of 3) + (Mr. Smith).
In math terms:
Number of men = 3a + 1, where a is a positive integer.
Options for the total number of men:
4, 7, 10, 13, 16, 19....
Only the value in green appears among the answer choices.

The correct answer is B.
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by Brent@GMATPrepNow » Tue Sep 04, 2018 7:09 am
vaibhav101 wrote:A group of people met at a party. Each person shook hands with everyone else. Mr. Smith shook hands with 3 times as many men as women. Mrs. Smith shook hands with 4 times as many men as women. How many men and women were there at the party?

A 5 , 16
B 16 , 5
C 18 , 3
D 12 , 9
E 9 , 12
Here's an algebraic solution...

Let M = TOTAL number of men (including Mr. Smith) at the party
Let W = TOTAL number of women (including Mrs. Smith) at the party

Since Mr. Smith does not shake hands with HIMSELF, Mr. Smith shakes hands with M-1 men, and he shakes hands with W women
Since Mrs. Smith does not shake hands with HERSELF, Mrs. Smith shakes hands with M men, and she shakes hands with W-1 women

Mr. Smith shook hands with 3 times as many men as women
We can write: M - 1 = 3W

Mrs. Smith shook hands with 4 times as many men as women.
We can write: M = 4(W - 1)

How many men and women were there at the party?
We now have 2 equations:
M - 1 = 3W
M = 4(W - 1)

Rewrite top equation to get:
M = 3W + 1
M = 4(W - 1)

Since both equations are set equal to M, we can write: 3W + 1 =4(W - 1)
Expand right side to get: 3W + 1 =4W - 4
Add 4 to both sides to get: 3W + 5 =4W
Subtract 3W from both sides to get: 5 =W

So, there are 5 women.
Check the answer choices.....only answer choice B has 5 women at the party.
Answer: B

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Brent
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by Scott@TargetTestPrep » Tue Oct 02, 2018 6:26 pm
vaibhav101 wrote:a group of people met at a party. Each person shook hands with everyone else. Mr. Smith shook hands with 3 times as many men as women. Mrs. Smith shook hands with 4 times as many men as women. How many men and women were there at the party?

A 5 , 16
B 16 , 5
C 18 , 3
D 12 , 9
E 9 , 12
Let m = the number of men at the party and w = the number of women at the party. Since Mr. Smith did not shake hands with himself, we have:

m - 1 = 3w

Likewise, since Mrs. Smith did not shake hand with herself, we have:

m = 4(w - 1)

Substituting 4(w - 1) for m in the first equation, we have:

4(w - 1) - 1 = 3w

4w - 4 - 1 = 3w

w = 5

So m = 4(5 - 1) = 16.

Alternate Solution:

Let's test each answer choice:

A) 5 men, 16 women

If there were 5 men and 16 women in the party, then Mr. Smith would have shaken hands with 5 - 1 = 4 men (since he doesn't shake his own hand) and 16 women. Since he didn't shake hands with 3 times as many men than women, this cannot be the correct answer choice.

B) 16 men, 5 women

If there were 16 men and 5 women in the party, Mr. Smith would have shaken hands with 16 - 1 = 15 men and 5 women, 3 times as many. Furthermore, Mrs. Smith would have shaken hands with 16 men and 5 - 1 = 4 women, 4 times as many. This is the correct answer choice.

Answer: B

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