"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."
let the number of men be x
the number of women be y
and the number of children be (100-x-y)
then 3x + 2y + (1/2)(100-x-y) = 100
6x + 4y + 100 - x - y = 200
5x + 3y = 100
this is a linear relations where one obvious solution is (20,0)
the 'slope' is -5/3, that is,
for every decrease of 3 in the x's we can increase y by 5 and we have another solution.
so other solutions are
(17,5)
(14,10)
(11,15) etc.
of course we have to check if the number of children is still valid.
e.g. for (11,15)
we would have 11 men, 15 women and 100-11-15 or 74 children
check: 11(3) + 15(2) + (1/2)(74) = 100 YEAH!
keep my pattern going for the rest of the results.
To distribute the 100 bushels of corn among 100 men, women, and children, we can use the given information that men receive 3 bushels each, women receive 2 bushels each, and children receive 1/2 bushel each.
Let's assign variables to represent the number of men, women, and children. We'll call the number of men x, the number of women y, and the number of children 100 - x - y (since there are 100 people in total).
Using this information, we can set up the equation 3x + 2y + (1/2)(100 - x - y) = 100 to represent the distribution of bushels. Simplifying this equation, we get 6x + 4y + 100 - x - y = 200, which can be further simplified to 5x + 3y = 100.
This linear equation represents the distribution of the bushels among the men, women, and children.
One solution to this equation is x = 20 (men) and y = 0 (women). This means that there are 20 men and 0 women. We can verify this solution by plugging in these values into the equation: 5(20) + 3(0) = 100, which is true.
However, there are also other solutions to this equation. To find these solutions, we can look at the slope of the equation, which is -5/3. This means that for every decrease of 3 in the x-values, we can increase the y-values by 5 and still satisfy the equation.
Using this pattern, we can find other solutions:
- For x = 17 (men), y = 5 (women), we have 17 men, 5 women, and 100 - 17 - 5 = 78 children. We can verify this solution by plugging in the values: 5(17) + 3(5) = 100, which is true.
- For x = 14 (men), y = 10 (women), we have 14 men, 10 women, and 100 - 14 - 10 = 76 children. We can verify this solution by plugging in the values: 5(14) + 3(10) = 100, which is true.
- For x = 11 (men), y = 15 (women), we have 11 men, 15 women, and 100 - 11 - 15 = 74 children. We can verify this solution by plugging in the values: 5(11) + 3(15) = 100, which is true.
By continuing this pattern, we can find more solutions to the equation. However, it's important to verify that the number of children is still valid in each solution.