in a group of 180 person composed of men, women and children there are twice as many men as women and three times as many women as children. how many are there each?
number of men --- m
number of women --- w
number of children --- c
m+w+c = 180 , #1
m = 2w , #2
w = 3c , #3
from #2 , w = m/2
then in #3,
3c = w
c = (1/3)w = (1/3)(m/2) = m/6
back in #1
m+w+c = 180
m + m/2 + m/6 = 180
times 6
6m + 3m + m = 1080
10m= 1080
m = 108
then w = m/2 = 54
c = m/6 = 18
108 men, 54 women, and 18 children
check:
are there twice as many men as women? YES
are there three times as many women as children? YE
Are there 180 in total? YES
All is good!
To solve this problem, we can use a system of equations. Let's assign variables to represent the unknowns:
Let's say the number of men is represented by M.
The number of women is represented by W.
The number of children is represented by C.
We are given three pieces of information:
1. "In a group of 180 persons composed of men, women, and children, there are twice as many men as women."
This can be written as: M = 2W.
2. "There are three times as many women as children."
This can be written as: W = 3C.
3. "The total number of people in the group is 180."
This can be written as: M + W + C = 180.
Using these three equations, we can solve for the unknowns.
Substituting the second equation into the first equation, we get:
M = 2(3C) = 6C.
We can substitute the expressions for M and W into the third equation:
6C + 3C + C = 180.
Combining like terms, we have:
10C = 180.
To isolate C, we divide both sides of the equation by 10:
C = 180/10 = 18.
Now that we know the value of C, we can substitute it back into the second equation to find W:
W = 3C = 3(18) = 54.
Finally, substitute the values of C and W into the first equation to find M:
M = 2W = 2(54) = 108.
Therefore, in the group of 180 persons, there are 108 men, 54 women, and 18 children.