A certain sum is divided among A, B and C in such a way that the ratio of shares of A and B is $3:4$ and that of shares of C and B is $9:5$. If the difference between the shares of A and C is ₹$462$, then the sum (in ₹) is:

Given the individual ratios: $A:B=3:4$$C:B=9:5⟹B:C=5:9$ To combine these into a single ratio $A:B:C$, make the value of $B$ the same in both ratios by multiplying by appropriate factors. The LCM of the values of B ($4$ and $5$) is $20$.Multiply $A:B=3:4$ by $5⟹A:B=15:20$ Multiply $B:C=5:9$ by $4⟹B:C=20:36$ Combining them gives:
$$A:B:C=15:20:36$$
Let the shares be $A=15x$, $B=20x$, and $C=36x$.Given that the difference between the shares of A and C is ₹$462$:
$$|C-A|=462$$
$$36x-15x=462$$
$$21x=462$$
$$x=\frac{462}{21}=22$$
The total sum is the sum of all shares:
$$\text{Total Sum}=A+B+C=15x+20x+36x=71x$$
$$\text{Total Sum}=71\times 22=1562$$
Thus, the total sum is ₹$1,562$.