A man lost half of his initial amount in the gambling after playing 3 rounds. The rule of gambling is that if he wins he will receive Rs. 100, but he has to give 50% of the total amount after each round. Luckily he won all the three rounds. The initial amount with which he had started the gambling was :

Options

A. a. `(500)/(3)`
B. b. `(700)/(3)`
C. c. 300
D. d. 600

Answer

To solve the problem step by step, we will denote the initial amount of money the man started with as \( X \).

### Step 1: Calculate the amount after the first round
When the man wins the first round, he receives Rs. 100. However, he has to give away 50% of his total amount after winning.

- After winning, his total amount becomes:
\[ \text{Total after first round} = X + 100 \]
- He then gives away 50% of this total:
\[ \text{Amount given away} = \frac{1}{2}(X + 100) \]
- Therefore, the amount left after the first round is:
\[ \text{Amount left} = (X + 100) - \frac{1}{2}(X + 100) = \frac{1}{2}(X + 100) \]

### Step 2: Calculate the amount after the second round
In the second round, he again wins Rs. 100. Now his total amount is:
\[ \text{Total after second round} = \frac{1}{2}(X + 100) + 100 \]
He gives away 50% of this total:
\[ \text{Amount given away} = \frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) \]
The amount left after the second round is:
\[ \text{Amount left} = \left(\frac{1}{2}(X + 100) + 100\right) - \frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) = \frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) \]

### Step 3: Calculate the amount after the third round
In the third round, he wins Rs. 100 again. His total amount now is:
\[ \text{Total after third round} = \frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) + 100 \]
He gives away 50% of this total:
\[ \text{Amount given away} = \frac{1}{2}\left(\frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) + 100\right) \]
The amount left after the third round is:
\[ \text{Amount left} = \left(\frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) + 100\right) - \frac{1}{2}\left(\frac{1}{2}\left(\frac{1}{2}(X + 100) + 100\right) + 100\right) \]

### Step 4: Set up the equation
According to the problem, after the three rounds, the man lost half of his initial amount. Therefore, the amount left after the third round should equal \( \frac{X}{2} \):
\[ \text{Amount left after third round} = \frac{X}{2} \]

### Step 5: Solve the equation
Now we can solve for \( X \):
1. Combine the expressions from the previous steps.
2. Set the final amount equal to \( \frac{X}{2} \) and solve for \( X \).

After simplifying the equations, we find:
\[ X = 700 \]

### Final Answer
The initial amount with which he started gambling was Rs. 700.

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