One says, "Give me a hundred, friend! I shall then become twice as rich as you." The other replies, "If you give me ten, I shall be six times as rich as you." Tell me what is the amount of their respective capital?

Given:

One says, "Give me a hundred, friend! I shall then become twice as rich as you."

The other replies, "If you give me ten, I shall be six times as rich as you."

To do:

We have to find the amount of their respective capital.

Let the capital with the first friend and the capital with the second friend be $x$ and $y$ respectively.

If the first friend gets 100 from the second friend then the money with first friend is $x+100$ and the money with the second friend is $y-100$.

If the second friend gets 10 from the first friend then the money with the first friend is $x-10$ and the money with the second friend is $y+10$.

According to the question,

$x + 100 = 2(y-100)$.....(i)

$y + 10 = 6(x-10)$

$y+10=6x-60$

$y=6x-60-10$

$y=6x-70$.....(ii)

Substituting $y=6x-70$ in equation (i), we get,

$x+100=2(6x-70-100)$

$x+100=12x-340$

$12x-x=100+340$

$11x=440$

$x=\frac{440}{11}$

$x=40$

This implies,

$y=6(40)-70$

$y=240-70$

$y=170$

The capital with the first friend is Rs. 40 and the capital with the second friend is Rs. 170.