Ajay, Bhavan and Chetan had a total of 150 marbles with them. Ajay gave each of Bhavan and Chetan as many marbles as they already had. Bhavan, then gave each of Ajay and Chetan as many marbles as they already had. If Bhavan finally has 90 marbles, find the number of marbles that he initially had.

This problem can be solved by working backwards from the final state of the marbles.

Step 1: We know that Bhavan finally has 90 marbles. Before he gave his marbles to Ajay and Chetan, he must have had twice this amount (since he gave each of them as many marbles as they already had). So, Bhavan had 90 * 2 = 180 marbles before he gave any away.

Step 2: Before Bhavan received any marbles from Ajay, he must have had half this amount (since Ajay gave him as many marbles as he already had). So, Bhavan initially had 180 / 2 = 90 marbles.

However, this answer is incorrect because it exceeds the total number of marbles (150). This indicates that the distribution of marbles among Ajay, Bhavan, and Chetan was not equal after Ajay gave away his marbles.

Let's try a different approach.

Step 1: Let's denote the initial number of marbles that Ajay, Bhavan, and Chetan had as A, B, and C respectively. After Ajay gave away his marbles, the number of marbles that Bhavan and Chetan had doubled. So, we have the equation: A = B + C.

Step 2: After Bhavan gave away his marbles, the number of marbles that Ajay and Chetan had doubled. Since Bhavan finally has 90 marbles, he must have had 90 * 2 = 180 marbles before he gave any away. So, we have the equation: B = 180 - A - C.

Step 3: We also know that the total number of marbles is 150. So, we have the equation: A + B + C = 150.

Now we can solve these three equations to find the initial number of marbles that Bhavan had.

From equation 1 and 2, we have: B = 180 - 2B. Solving this equation gives B = 60.

So, Bhavan initially had 60 marbles.