A person divided a certain sum among his three sons in the ratio 2 : 3 : 8. If he had instead divided it in the ratio $$\frac{1}{2} : \frac{1}{3} : \frac{1}{8}$$, the son who got the least share would have received ₹2200 more. The sum (in ₹) was:

A person divided a certain sum among his three sons in the ratio 2 : 3 : 8.

Let's assume that the initial amount is 13y and sons got 2y, 3y and 8y respectively.
If he had instead divided it in the ratio $$\frac{1}{2} : \frac{1}{3} : \frac{1}{8}$$.

$$\frac{12}{24}:\frac{8}{24}:\frac{3}{24}$$

So the new ratio are 12 : 8 : 3.

Let's assume 12z, 8z and 3z. Total amount = 12z+8z+3z = 23z.

We know that the amount was the same. So 13y = 23z.

$$\frac{y}{z}\ =\ \frac{23}{13}$$

So let's assume y = 23a and z = 13a.    Eq.(i)

The son who got the least share would have received ₹2200 more.

2y+2200 = 12z

y+1100 = 6z

Put Eq.(i) in the above equation.

$$23a+1100 = 6\times13a$$

$$23a+1100 = 78a$$

78a-23a = 1100

55a = 1100

a = 20    Eq.(i)

So the total amount = 13y

= $$13\times23a$$                                     (From Eq.(i).)

= $$13\times23\times20$$                          (From Eq.(ii).)

= ₹5980