A daily wage worker works for 23 days in a month. He gets Rs 400 per working day. However, his supervisor is very strict and fines him Rs 100 for each day he arrives late to work. If at the end of the 23 days, the worker has earned Rs 8000, find out the number of days he reached for work on time and the number of days he was late.

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = number of days the worker arrives on time
- \( y \) = number of days the worker is late

### Step 2: Set Up the First Equation
According to the problem, the worker works for a total of 23 days in a month. Therefore, we can write the first equation as:
\[ x + y = 23 \]

### Step 3: Express \( y \) in Terms of \( x \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 23 - x \]

### Step 4: Calculate Earnings
The worker earns Rs 400 for each day he works on time. For the days he is late, he earns Rs 300 (since he is fined Rs 100). Therefore, his total earnings can be expressed as:
\[ \text{Total Earnings} = 400x + 300y \]

### Step 5: Set Up the Second Equation
According to the problem, the total earnings after 23 days is Rs 8000. Thus, we can write the second equation as:
\[ 400x + 300y = 8000 \]

### Step 6: Substitute \( y \) in the Second Equation
Now, substitute \( y \) from Step 3 into the second equation:
\[ 400x + 300(23 - x) = 8000 \]

### Step 7: Simplify the Equation
Now simplify the equation:
\[ 400x + 6900 - 300x = 8000 \]
Combine like terms:
\[ 100x + 6900 = 8000 \]

### Step 8: Solve for \( x \)
Now, isolate \( x \):
\[ 100x = 8000 - 6900 \]
\[ 100x = 1100 \]
\[ x = \frac{1100}{100} \]
\[ x = 11 \]

### Step 9: Calculate \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = 23 - x \]
\[ y = 23 - 11 \]
\[ y = 12 \]

### Final Answer
The worker was on time for **11 days** and was late for **12 days**.

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