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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

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Oct 13, 2011
The Village and 100 Loaves of Bread
by: Staff

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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)
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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


Oct 13, 2011
The Village and 100 Loaves of Bread
by: Staff


Part I

Question:

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.
Answer:

You can organize all this information using the five steps:


The five steps are:

1. Understand what the problem is asking you to do
2. Identify the information provided (known and unknown information)
3. Write (or rewrite) the known and unknown information using mathematical notation (rather than sentences)
4. Solve the problem
5. Check your answer



Applying these 5 steps to your problem:

1. Understand what the problem is asking you to do

(Read the last sentence in the problem statement.)

How many men, women and children lived in the village?

Find all 7 possible answers to this question.


2. Identify the information provided (known and unknown information)

Known Information

- The small village had a population of exactly 100 people.

- The truck contained 100 loaves of bread.

- each child will receive half a loaf of bread

- each woman will receive two loaves of bread

- each man three



Unknown Information

- the number of men

- the number of women

- the number of children



3. Write (or rewrite) the known and unknown information using mathematical notation (rather than sentences)

B = loaves of bread = 100

M = number of men = unknown

W = number of women = unknown

C = number of children = unknown


the number of people living in the village is:

M + W + C = 100


the number of loaves distributed is:

M * 3 + W * 2 + C * (1/2) = 100

or

3M + 2W + (1/2)C = 100

This will be easier to work with if the fraction ½ is removed. The remove the fraction, multiply each side of the equation by 2

2 * [3M + 2W + (1/2)C] = 2 * 100

6M + 4W +C = 200



4. Solve the equations for M, W, and C

As you can see, there are three unknowns, but only 2 equations.

1st equation: M + W + C = 100

2nd equation: 6M + 4W + C = 200

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