A father wishes to divide his 500-acre farm between his three sons in portions of land directly proportional to the number of kids each one has and inversely proportional to their incomes. If the oldest son's income is twice the youngest's and the middle son's income is thrice the youngest's, and if the oldest son has three kids, the middle son has two kids and the youngest son has two kids, how many acres of land will the middle son receive?
a) 80; b) 100; c) 120; d) 160; e) 180
Here's my attempt:
By the definitions of directly and inversely proportional division, we have
$$\frac{2k}{x}+\frac{2k}{3x}+\frac{3k}{2x}=500$$
where $x$ is he middle son's income and $k$, the constant of proportionality.
But I'm not sure how to proceed from here. Shouldn't the portion of land that is due to each son be a function of $x$? Any help will be appreciated.
Suppose the portion of land allocated to the youngest is denoted $L$.
Now, we have that land allocation is proportional to number of kids, and inversely proportional to income.
We know that the middle son has the same number of kids as the youngest. Therefore, his allocation of land is $\frac{1}{3}L$. Now, the oldest son's income is twice the youngest, but he has three kids whilst the youngest has two. Therefore he gets $\frac{1}{2}\frac{3}{2}L = \frac{3}{4}L$.
Now we have that there are $500$ acres going around altogether, so that $\frac{25}{12}L = L + \frac{1}{3}L + \frac{3}{4}L = 500$.
From this you can solve for $L$, and figure out which option the middle son got, with his $\frac{1}{3}L$.