Problem Solving

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