Question

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? [From the Bijaganita of Bhaskara II]

Answer 1

Let the initial amount with them be $\text{Rs.}x$ and $\text{Rs.}y$ respectively.

Then according to the question:

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

$x+100=2y-200$

$x-2y=-300\dots \left(1\right)$

And,

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

$6x-60=y+10$

$6x-y=70\dots \left(2\right)$

Multiplying equation $\left(2\right)$ by 2, we get:

$12x-2y=140\dots \left(3\right)$

Subtracting equation $\left(1\right)$ from equation $\left(3\right)$, we get:

$11x=140+300$

$11x=440$

$x=40$

Substitute the vale of $x$ in equation $\left(1\right)$, we get:

$40-2y=-300$

$40+300=2y$

$2y=340$

$y=\frac{340}{2}=170$

$\therefore$ The friends had $\text{Rs.}40$ and $\text{Rs.}170$ with them respectively.