there are 7 possible answers find three?

Question

the relief truck arrived at the village to distribute much needed food. The small village had a population of exaclty 100 people. The truck contained 100 lovaes of bread, and the food was about to be distributed evenly when the village chieftain suddenly stopped proceedings. "the elders of the village have decided that each child will recieve half a loaf of bread, each woman two loaves of bread and each mand three," bellowed the chieftain/ " But we may not have enough bread in those proportions," argued a relief worker. :You have 100 loaves?' asked the chieftain. "yes" replied the worker. "then thats exactly enough" How many men, women, and children lived in the village?

Answer 1

Let c = # of children
Let w = # of women
Let m = # of men

The original proportions were to be:
c + w + m = 100

But then the chieftan changes the proportions:
0.5c + 2w + 3m = 100

Subtract the two equations and you get:
-0.5c + w + 2m = 0
0.5c = w + 2m
c = 2w + 4m

Now replace this value of c into the original equation:
2w + 4m + w + m = 100

You now have a ratio of men to women.
3w + 5m = 100

Now try some numbers:
m = 1 --> fractional amount of women
m = 2, w = 30, c = 68 (6 loaves + 60 loaves + 34 loaves)
m = 3 or 4 --> fractional amount of women
m = 5, w = 25, c = 70 (15 loaves + 50 loaves + 35 loaves)
m = 6 or 7 --> fraction amount of women
m = 8, w = 20, c = 72 (24 loaves + 40 loaves + 36 loaves)

There are 3 answers:
m = 2, w = 30, c = 68 (6 loaves + 60 loaves + 34 loaves)
m = 5, w = 25, c = 70 (15 loaves + 50 loaves + 35 loaves)
m = 8, w = 20, c = 72 (24 loaves + 40 loaves + 36 loaves)

The remaining answers are:
m = 11, w = 15, c = 74
m = 14, w = 10, c = 76
m = 17, w = 5, c = 78
m = 20, w = 0, c = 80

Answer 2

Let c be the number of children, and w be the number of women. Then the number of men is 100 - c - w.

Each child got half a loaf, each woman two, and each man three. So these should all add up to 100 loaves.
(1/2)c + 2w + 3(100-c-w) = 100

This simplifies to
(1/2)c + 2w + 300 - 3c - 3w = 100
(1/2)c + 2w + 200 - 3c - 3w = 0
(-5/2)c - w + 200 = 0
(5/2)c + w = 200
5c + 2w = 400

Try different integer solutions. Since 2w is even and 400 is even, 5c has to be even too which means c has to be even. Also, since 5c is a multiple of 5 and so is 400, then so must be 2w. So w must be a multiple of 5. But 100 - c - w still has to be positive. Try w=5, 10, 15, etc. and solve for c to get some answers.

Answer 3

100 :)