Question 249145: If 100 bushels of corn is distributed among 100 people in such a manner that each man receives 3 bushels, each woman 2, and each child 1/2 of a bushel, which one of the following is a possible value for the number of men ?
(A)16 (B) 12 (C) 8 (D) 4 (E) 0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I would have to say selection (C) which is equal to 8.
Here's why:
you have 2 equations in 3 unknowns which means the best you can do is find a relationship between 2 of the variables.
Let:
x = number of men
y = number of women
z = number of children
your equations are:
x + y + z = 100
3x + 2y + z/2 = 100
you can eliminate one of the variable by multiplying the second equation by 2 to get:
x + y + z = 100
minus: 6x + 4y + z = 200
equals: -5x - 3y = -100
you can solve for y in the resulting equation as follows:
your equation to solve is:
-5x - 3y = -100
add 5x to both sides to get:
-3y = -100 + 5x
multiply both sides by -1 to get:
3y = 100 - 5x
divide both sides by 3 to get:
y = (100-5x)/3
you equation you need to work with is:
y = (100-5x)/3
look at all your solutions and solve for the given value of x.
when x = 16, y = 6.67
when x = 12, y = 13.33
when x = 8, y = 20
when x = 4, y = 26.67
when x = 0, y = 33.33
in all of these, the only integer value is 20, so the only possible solution would be x = 8 and y = 20
that means that there could be 8 men and 20 women which means that there must be 72 children because x + y + z = 100
if so, then the distribution is among 8 men, 20 women, and 72 children.
each man gets 3 bushels so the men get 24 bushels
each woman gets 2 bushels so the women get 40 bushels
each child gets 1/2 bushel so the children get 36 bushels.
24 + 40 + 36 = 100 bushels so that checks out ok.
your answer is selection (C) which is 8 men as a possible solution.
there could be other solutions as well since you did not solve for the number of women.
you only solved for the number of women in relataion to the number of men.
example:
if x = 2, then (100-5x)/3 = 90/3 = 30 women.
you could have 2 men, 30 women, and 68 children
3*2 = 6
2 * 30 = 60
68 * .5 = 34
60 + 34 + 6 = 100
number of men equals 2 is a possibility as well.
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