A total fixed amount of $$N$$ thousand rupees is given to three persons $$A$$, $$B$$, $$C$$, every year, each being given an amount proportional to her age. In the first year, $$A$$ got half the total amount. When the sixth payment was made, $$A$$ got six-seventh of the amount that she had in the first year; $$B$$ got Rs 1000 less than that she had in the first year; and $$C$$ got twice of that she had in the first year. Find $$N$$.
Correct Answer: 35
Solution
Let the first year ages be $$a, b, c$$. Since $$A$$ received half the total, $$a = b+c$$, so the total age is $$2a$$ and six years later it is $$2a+15$$. The condition on $$A$$ gives $$\frac{a+5}{2a+15} = \frac{3}{7}$$, so $$a = 10$$ and $$b+c = 10$$, and the condition on $$C$$ gives $$\frac{c+5}{35} = \frac{c}{10}$$, so $$c = 2$$ and $$b = 8$$. Finally $$B$$ gives $$\frac{13N}{35} = \frac{2N}{5} - 1$$, which yields $$N = 35$$.
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