Question: If Prakash gives one of the marbles from what he posseses to Kamala, then they will have equal number or marbles. If Kamala gives one of the marbles from what she posseses to Prakash, then Prakash will have double number of marbles. Find the number of marbles posseses by them initially.

Solution:

Let the number of marbles that Prakash possesses be x and the number of marbles that Kamala possesses be y.

According to the question,

Condition I,
If Prakash gives one of the marbles from what he posseses to Kamala, then they will have equal number of marbles.
$or, x -1 = y +1$
$or, x = y +1+1$
$or, x = y +2$ - (i)

Condition II,
If Kamala gives one of the marbles from what she possesses to Prakash, then Prakash will have double number of marbles.
$or, (x +1) = 2(y -1)$
$or, x + 1 = 2y -2$
$or, x = 2y -2-1$
$or, x = 2y -3$ - (ii)

Put value of x from equation (i) in equation (ii), we get,

$or, y +2 = 2y -3$
$or, 2+3 = 2y -y$
$or, 5 = y$
$\therefore y = 5$

Put value of y in equation (i), we get,

$or, x = 5 +2$
$\therefore x = 7$

So, (x,y) = (7,5)

Therefore, Prakash had 7 marbles and Kamala had 5 marbles with them.


Explanation:
If Prakash gives one of the marble to Kamala, then the marbles that Prakash possesses is decreased by 1. Also, the marble that Kamala possesses increases by 1.

If Prakash has 7 marbles and Kamala has 5. When Prakash gives one marble to Kamala, he will have 6 marbles which is 7-1. Also, Kamala will have 6 marbles that is 5+1.

As a result, when one marble is given by Prakash to Kamala,
x - 1 = y +1, where x refers to the marbles that Prakash has and y refers to the marbles that Kamala has.