Answer to Question #20121 in Abstract Algebra for Alicia Dunigan

Question

24 people were traveling and brought 24 loaves of bread. Each man brought 2 loaves, each woman brought half a loaf, and each child brought a quarter of a loaf. How many men were traveling? How many women were traveling? How many children were traveling?

Answer 1

Let's denote the number brought by men, women and children by m, w and c respectfully. Then we can write down the following system of equations:

2m + w/2 + c/4 = 24 (1)
m + w + c = 24& (2)
mєN, wєN, cєN.

Let's solve it. Subtracting equation (1) from doubled equation (2) we obtain:

1.5w + 1.75c = 24 (3)

w and c have to be integers, and (3) can only be satisfied if w = 2 and c = 12. Therefore,

m = 24 - c/4 - w/2 = 24 - 12/4 - 2/2 = 20.

So, there were travelling 20 men, 2 women and 12 children.