  # The Village and 100 Loaves of Bread

by Andrew
(New York)

The relief truck arrived at the village to distribute much-needed food. The small village had a population of exactly 100 people. The truck contained 100 loaves of bread, and the food was about to be distributed evenly when the village chieftain suddenly stopped the proceedings.
“The elders of the village have decided that each child will receive half a loaf of bread, each woman two loaves and each man 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 that is exactly enough.”
How many men, women and children lived in the village?
Find all 7 possible answers to this question.

### Comments for The Village and 100 Loaves of Bread

 Oct 13, 2011 The Village and 100 Loaves of Bread by: Staff ---------------------------------------------- Part II Therefore, this problem will ultimately be solved by trial and error. Subtract the 1st equation from the 2nd equation to eliminate the variable C 6M + 4W + C = 200 -( M + W + C = 100) -------------------------- 6M - M + 4W - W + C - C = 200 - 100 (6M – M) + (4W – W) + (C – C) = (200 – 100) 5M + 3W + 0 = 100 5M + 3W = 100 Solve for M (or W, if you prefer) 5M + 3W - 3W = 100 - 3W 5M + (3W - 3W) = 100 - 3W 5M + 0 = 100 - 3W 5M = 100 - 3W 5M = 100 - 3W 5M / 5= (100 - 3W) / 5 M * (5 / 5)= (100 - 3W) / 5 M * (1)= (100 - 3W) / 5 M = (200 - 3W) / 5 M = (100 / 5) - (3W / 5) M = 20 - (3 / 5)W M = 20 - 0.6W The number of men and the number of women must be whole numbers, since there cannot be 0.6 of a woman. Since 0.6 * W must be a whole number, W can only be multiples of 5: W = 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, . . . 100 If W must be a 0, 5 or multiples of 5, possible values of M can be calculated: If you compute all possible values of W and M, this is easy to see W . . M = 20 - 0.6W 0 . . . 20 1 . . . 19.4 2 . . . 18.8 3 . . . 18.2 4 . . . 17.6 5 . . . 17 6 . . . 16.4 7 . . . 15.8 8 . . . 15.2 9 . . . 14.6 10 . . 14 11 . . 13.4 12 . . 12.8 13 . . 12.2 14 . . 11.6 15 . . 11 16 . . 10.4 17 . . 9.8 18 . . 9.2 19 . . 8.6 20 . . 8 21 . . 7.4 22 . . 6.8 23 . . 6.2 24 . . 5.6 25 . . 5 26 . . 4.4 27 . . 3.8 28 . . 3.2 29 . . 2.6 30 . . 2 31 . . 1.4 32 . . 0.8 33 . . 0.2 34 . . -0.4 (M can never be negative) Since both W and M must be positive integers (whole people) ≥ 0, the only possibilities are: W . . M 0 . . . 20 5 . . . 17 10 . . 14 15 . . 11 20 . . 8 25 . . 5 30 . . 2 Based on these possibilities, it is now possible to calculate the number of children for each set of W and M. The number of children C must also be an integer ≥ 0. M + W + C = 100 Therefore: C = 100 - M – W The only possibilities for W, M, and C are W . . M . . C = 100 - M - W 0 . . . 20 . . 80 5 . . . 17 . . 78 10 . . 14 . . 76 15 . . 11 . . 74 20 . . 8 . . . 72 25 . . 5 . . . 70 30 . . 2 . . . 68 The final answer is the number of women, number of men, the number of children 0 . . . 20 . . 80 5 . . . 17 . . 78 10 . . 14 . . 76 15 . . 11 . . 74 20 . . 8 . . . 72 25 . . 5 . . . 70 30 . . 2 . . . 68 5. Check the answer The total number of loaves of bread must = 100. B = 3M + 2W + (1/2)C W . . M . . C . . B = 3M + 2W + (1/2)C 0 . . . 20 . . 80 . . 100 5 . . . 17 . . 78 . . 100 10 . . 14 . . 76 . . 100 15 . . 11 . . 74 . . 100 20 . . 8 . . . 72 . . 100 25 . . 5 . . . 70 . . 100 30 . . 2 . . . 68 . . 100 Thanks for writing. Staff www.solving-math-problems.com