A man plays a gambling game in which, in each round he wins Rs. 100 but must give away 50% of his total amount at the end of that round. He plays three rounds and wins all three games. After these rounds he has lost exactly half of his initial amount. What was the initial amount of money with which he started the gambling?

Introduction / Context:
This is a multi step percentage and algebra puzzle that tests careful reading and systematic calculation. In each round of the gambling game, the player first wins a fixed sum of money and then must give away half of his total amount. After three such rounds, his final amount is exactly half of his original amount. From this information, you must work backwards to recover the initial amount of money.

Given Data / Assumptions:

The man starts with some initial amount of money, say X rupees.

In each round he wins Rs. 100.

At the end of every round he must give away 50% of his total amount at that moment.

He plays 3 rounds and wins all 3.

After the three rounds he has lost half of his initial amount, so his final amount equals X / 2.

Concept / Approach:
We treat this as a stepwise process. Let the amount after round 1 be A1, after round 2 be A2, and after round 3 be A3. For each round, the sequence is: add Rs. 100 to the current amount, then halve it. So A1 = (X + 100) / 2, A2 = (A1 + 100) / 2, and A3 = (A2 + 100) / 2. We then set A3 equal to X / 2 and solve for X using algebra. This gives a linear equation in X.

Step-by-Step Solution:
Step 1: Let initial amount = X rupees.Step 2: After round 1: he wins 100, so amount becomes X + 100. He then gives away half, so A1 = (X + 100) / 2.Step 3: After round 2: starting from A1, he wins 100, new amount = A1 + 100, then gives half, so A2 = (A1 + 100) / 2.Step 4: After round 3: starting from A2, he wins 100, new amount = A2 + 100, then gives half, so A3 = (A2 + 100) / 2.Step 5: We are told A3 equals half of the initial amount, so A3 = X / 2.Step 6: Substitute A1 and A2 step by step. First A1 = (X + 100) / 2.A2 = (A1 + 100) / 2 = ((X + 100) / 2 + 100) / 2.A3 = (A2 + 100) / 2.After simplifying this expression, we obtain A3 = X / 8 + 175 / 2.Step 7: Set X / 8 + 175 / 2 equal to X / 2.So X / 8 + 175 / 2 = X / 2.Step 8: Multiply through by 8 to clear denominators: X + 700 = 4X.Step 9: Rearrange: 4X − X = 700, so 3X = 700 and X = 700 / 3.

Verification / Alternative check:
Calculate the amounts numerically starting from X = 700 / 3. Round 1: X + 100 = 700 / 3 + 100 = 1000 / 3. After giving half: A1 = 1000 / 3 / 2 = 500 / 3. Round 2: A1 + 100 = 500 / 3 + 100 = 800 / 3. After giving half: A2 = 800 / 3 / 2 = 400 / 3. Round 3: A2 + 100 = 400 / 3 + 100 = 700 / 3. After giving half: A3 = 700 / 3 / 2 = 350 / 3. Since initial amount X = 700 / 3, half of X is 350 / 3, which equals A3. This confirms the solution.

Why Other Options Are Wrong:
500/3, 300, 400, and 600 do not satisfy the relation that the final amount after three rounds equals half of the starting amount when you simulate the process step by step.

Common Pitfalls:
Common errors include halving before adding 100 instead of after, or applying 50% to the wrong intermediate amount. Some candidates also simplify too early or incorrectly handle fractions, leading to algebraic mistakes. The safest method is to write expressions carefully for each round, check the recursion, and then solve the final equation systematically.

Final Answer:
The initial amount of money was 700/3 rupees.