A man loses `20 1/2 %`of his money and after spending 80% of the remainder, he is left with Rs. 159. How much did he have at first?
एक आदमी को अपने धन का `20 1/2` % का नुकसान हो जाता है और शेष का 80% खर्च करने के बाद उसके पास Rs. 159 बचते हैं। शुरू में उसके पास कितने रुपये थे?

Options

A. Rs. 1200
B. Rs. 500
C. Rs. 800
D. Rs. 1000

Explanation

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

### Step 1: Calculate the amount lost
The man loses \( 20 \frac{1}{2} \% \) of his money. This can be expressed as a fraction:
\[ 20 \frac{1}{2} \% = \frac{41}{2} \% \]
To convert this percentage into a decimal for calculations, we divide by 100:
\[ \frac{41}{2} \% = \frac{41}{2} \times \frac{1}{100} = \frac{41}{200} \]
Now, the amount lost is:
\[ \text{Amount lost} = x \times \frac{41}{200} = \frac{41x}{200} \]

### Step 2: Calculate the remaining amount after the loss
The remaining amount after the loss can be calculated as:
\[ \text{Remaining amount} = x - \text{Amount lost} = x - \frac{41x}{200} \]
To simplify this, we need a common denominator:
\[ \text{Remaining amount} = \frac{200x}{200} - \frac{41x}{200} = \frac{200x - 41x}{200} = \frac{159x}{200} \]

### Step 3: Calculate the amount spent
The man spends \( 80\% \) of the remaining amount. Therefore, the amount spent is:
\[ \text{Amount spent} = 80\% \text{ of } \frac{159x}{200} = \frac{80}{100} \times \frac{159x}{200} = \frac{80 \times 159x}{20000} = \frac{12720x}{20000} \]

### Step 4: Calculate the remaining amount after spending
After spending \( 80\% \), he is left with \( 20\% \) of the remaining amount:
\[ \text{Amount left} = 20\% \text{ of } \frac{159x}{200} = \frac{20}{100} \times \frac{159x}{200} = \frac{159x}{1000} \]

### Step 5: Set up the equation
According to the problem, the amount left is Rs. 159:
\[ \frac{159x}{1000} = 159 \]

### Step 6: Solve for \( x \)
To find \( x \), we can multiply both sides by 1000:
\[ 159x = 159 \times 1000 \]
Now, divide both sides by 159:
\[ x = 1000 \]

### Conclusion
The initial amount of money the man had is Rs. 1000.