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
Cheers,
Brent
