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)

[**Hint:** *x* + 100 = 2 (*y* − 100), *y* + 10 = 6(*x* − 10)]

#### Solution

Let those friends were having Rs x and y with them.

Using the information given in the question, we obtain

*x* + 100 = 2(*y* − 100)

*x* + 100 = 2*y* − 200

*x* − 2*y* = −300 (*i*)

And, 6(*x* − 10) = (*y* + 10)

6*x* − 60 = *y* + 10

6*x* − *y* = 70 (*ii*)

Multiplying equation (ii) by 2, we obtain

12*x* − 2*y* = 140 (*iii*)

Subtracting equation (i) from equation (iii), we obtain

11*x* = 140 + 300

11*x* = 440

*x* = 40

Using this in equation (*i*), we obtain

40 − 2*y* = −300

40 + 300 = 2*y*

2*y* = 340

*y* = 170

Therefore, those friends had Rs 40 and Rs 170 with them respectively.