A person has some amount with him. `25% ` of its is stolen in a bus, `10%` is lost through a hole in the pocket, `50% ` of remainder is spent on food. He then purchases a book worth Rs. 26 from the remainder. He walks back home because all his money is over. What was the initial amount?

Answer

To solve the problem step by step, let's denote the initial amount of money the person has as \( x \).

### Step 1: Calculate the amount stolen
The person loses \( 25\% \) of the initial amount due to theft.
\[ \text{Amount stolen} = 25\% \text{ of } x = \frac{25}{100} \times x = 0.25x \]

### Step 2: Calculate the remaining amount after theft
After the theft, the remaining amount is:
\[ \text{Remaining amount after theft} = x - 0.25x = 0.75x \]

### Step 3: Calculate the amount lost through the hole
Next, the person loses \( 10\% \) of the remaining amount through a hole in the pocket.
\[ \text{Amount lost} = 10\% \text{ of } 0.75x = \frac{10}{100} \times 0.75x = 0.075x \]

### Step 4: Calculate the remaining amount after the loss
After this loss, the remaining amount is:
\[ \text{Remaining amount after loss} = 0.75x - 0.075x = 0.675x \]

### Step 5: Calculate the amount spent on food
The person spends \( 50\% \) of the remaining amount on food.
\[ \text{Amount spent on food} = 50\% \text{ of } 0.675x = \frac{50}{100} \times 0.675x = 0.3375x \]

### Step 6: Calculate the remaining amount after food expenses
After spending on food, the remaining amount is:
\[ \text{Remaining amount after food} = 0.675x - 0.3375x = 0.3375x \]

### Step 7: Calculate the amount left after buying the book
The person then buys a book worth Rs. 26 from the remaining amount. Therefore, we have:
\[ 0.3375x - 26 = 0 \]
This implies:
\[ 0.3375x = 26 \]

### Step 8: Solve for \( x \)
To find \( x \), we divide both sides by \( 0.3375 \):
\[ x = \frac{26}{0.3375} \]
Calculating this gives:
\[ x = 77.0 \text{ (approximately)} \]

### Step 9: Calculate the initial amount
To find the initial amount, we multiply \( x \) by 100:
\[ \text{Initial amount} = 100 \times 0.77 \approx 80 \]

Thus, the initial amount the person had is Rs. 80.

### Summary of the Solution
The initial amount was Rs. 80.

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