A total fixed amount of N thousand rupees is given to three persons...

Let total amount given each year be $N_1$, and the respective ages of persons $A, B$ and $C$ are $a, b$ and $c$ years respectively. $\therefore$ The ratio of initial distribution $=\mathrm{a}: \mathrm{b}: \mathrm{c}$. If $k$ is the ratio coefficient then $a k+b k+c k=N_1 \Rightarrow k=\frac{N_1}{a+b+c}$ A.T.Q: $\frac{a N_1}{a+b+c}=\frac{N_1}{2} \Rightarrow \frac{a}{a+b+c}=\frac{1}{2}$...(i)$and \frac{b+c}{a+b+c}=\frac{1}{2}$...(ii) When $6^{\text {th }}$ payment was made then ratio of distribution is $a+5: b+5: c+5$. $\therefore(a+5) k_1+(b+5) k_1+(c+5) k_1=N$ $\therefore k_1=\frac{N_1}{a+b+c+15}$ Again A.T.Q: $\frac{N_1(a+5)}{a+b+c+15}=\frac{6}{7} \cdot \frac{\mathrm{~N}}{2}$...(iii) $\frac{N_1(b+5)}{a+b+c+15}=\frac{b N_1}{a+b+c}-1000$...(iv) $\text { and } \frac{N_1(c+5)}{a+b+c+15}=\frac{2 c N_1}{a+b+c}$...(v) From equation (i) and (iii) $\frac{a+5}{2 a+15}=\frac{3}{7} \Rightarrow a=10 .$ From (ii) we get $b+c=10$ $\therefore a+b+c=20$ From equation (v) we get : $c=2$ then $b=8$. and from equation (iv), we get $$\begin{aligned} & \frac{13 N_1}{35}=\frac{8 N_1}{20}-1000 \Rightarrow N_1=35000 \\ & \therefore N=35 \text { thousand }\end{aligned}$$