100 bushels of corn are divided among 100 men, women, and children. Men receive 3 bushels each, women 2 bushels and children 1/2 bushel each. How can the bushels be distributed? is there more than one solution? if so, find the other solutions.
To solve this problem, we can break it down into three parts: the number of men, the number of women, and the number of children. Let's assume there are M men, W women, and C children.
Given that there are 100 bushels of corn, we can write the equation:
3M + 2W + 0.5C = 100
Since there are 100 people in total, we can also write:
M + W + C = 100
Now let's solve these equations to find the values of M, W, and C.
First, let's eliminate the fractions by multiplying the entire equation 3M + 2W + 0.5C = 100 by 2:
6M + 4W + C = 200
Now, let's subtract the second equation (M + W + C = 100) from this equation to eliminate C:
6M + 4W + C - M - W - C = 200 - 100
5M + 3W = 100
We now have two equations:
5M + 3W = 100
M + W + C = 100
To find the possible solutions, we can use trial and error. Start with different values of M (e.g., M = 0, 1, 2, 3...) and solve for W using the first equation. If M and W are integers, check if the corresponding value of C is also an integer. If it is, you have found a valid solution. Repeat this process until you find all possible solutions.
For example, let's try M = 0:
5(0) + 3W = 100
3W = 100
W = 100/3
Since W is not an integer, this is not a valid solution. Let's try other values of M.
By trying different values of M, W, and C, we can find the possible combinations that satisfy both equations. It is possible that there is more than one solution, but it would require further exploration to find all of them.