A man died. Let's divide the estate!!! How? This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with plain old algebra, which yields the shortest, simplest answers, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes: > A man dies and leave his estate to his sons. > > The estate is divided as follows: > > $1$st son gets 100 crowns $+$ $\frac{1}{10}$ of remainder of estate. > > $2$nd son gets 200 crowns $+$ $\frac{1}{10}$ of remainder of estate ... > > $(n)$$th$ son gets $100 × (n)$ crowns + $\frac{1}{10}$ of the remainder. > > Each son receives same amount. > > How many sons were there, what did each receive, and what was the estate?

Assuming all of his property was ' _crowns_ ', let the total amount he had be $\mathbf{T}$ crowns. Now the first son gets $\mathbf{100}$ crowns, so $\mathbf{T-100}$ crowns are left, of which he gets one-tenth, so his total inheritance is $\mathbf{\frac{T-100}{10} + 100}$ crowns. The second son first takes $\mathbf{200}$ crowns from the remaining estate, leaving us with $\mathbf{T-(200 + 100 + \frac{T-100}{10}) = \frac{9T - 2900}{10}}$ crowns, of which he takes one-tenth again. Gievn that each son receives an equal share, we equate these amounts with $\mathbf T$ being unknown $$\mathbf{\frac{T-100}{10} + 100 = \frac{9T - 2900}{100} + 200}$$ On solving, you are left with $\mathbf{T = 8100}$, so each son receives $\mathbf{900}$ crowns, i.e. there are $\mathbf9$ sons