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?

(A) Let the amounts of capital with the two friends be $Rs$ $x$ and $Rs$ $y$ respectively.
According to the first condition: "Give me a hundred,friend! $I$ shall then become twice as rich as you."
$x + 100 = 2(y - 100)$
$x + 100 = 2y - 200$
$x - 2y = -300$ $...(i)$
According to the second condition: "If you give me ten,$I$ shall be six times as rich as you."
$6(x - 10) = y + 10$
$6x - 60 = y + 10$
$6x - y = 70$ $...(ii)$
To solve the system,multiply equation $(ii)$ by $2$:
$12x - 2y = 140$ $...(iii)$
Subtract equation $(i)$ from equation $(iii)$:
$(12x - 2y) - (x - 2y) = 140 - (-300)$
$11x = 440$
$x = 40$
Substitute $x = 40$ into equation $(i)$:
$40 - 2y = -300$
$-2y = -340$
$y = 170$
Thus,the amounts of their respective capital are $Rs$ $40$ and $Rs$ $170$.