Some students planned a picnic. The total budget for food was Rs. 2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by Rs. 20. How many students attended the picnic and how much did each student pay for the food?

Explanation

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

### Step 1: Define the Variables
Let \( X \) be the total number of students who initially planned to attend the picnic.

### Step 2: Set Up the Initial Cost Per Head
The total budget for food is Rs. 2000. Therefore, the cost per head when all \( X \) students planned to attend is:
\[ \text{Cost per head} = \frac{2000}{X} \]

### Step 3: Set Up the Cost Per Head After Some Students Drop Out
Since 5 students failed to attend, the number of students who actually attended the picnic is \( X - 5 \). The new cost per head becomes:
\[ \text{New Cost per head} = \frac{2000}{X - 5} \]

### Step 4: Set Up the Equation Based on the Increase in Cost
According to the problem, the new cost per head is Rs. 20 more than the initial cost per head. Therefore, we can set up the equation:
\[ \frac{2000}{X - 5} - \frac{2000}{X} = 20 \]

### Step 5: Simplify the Equation
To solve this equation, we will first take 2000 common:
\[ 2000 \left( \frac{1}{X - 5} - \frac{1}{X} \right) = 20 \]
Dividing both sides by 2000 gives:
\[ \frac{1}{X - 5} - \frac{1}{X} = \frac{1}{100} \]

### Step 6: Find a Common Denominator
The common denominator for the left side is \( X(X - 5) \):
\[ \frac{X - (X - 5)}{X(X - 5)} = \frac{5}{X(X - 5)} = \frac{1}{100} \]

### Step 7: Cross-Multiply
Cross-multiplying gives:
\[ 5 \cdot 100 = X(X - 5) \]
This simplifies to:
\[ 500 = X^2 - 5X \]

### Step 8: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ X^2 - 5X - 500 = 0 \]

### Step 9: Factor the Quadratic Equation
We will factor the quadratic equation:
\[ X^2 - 25X + 20X - 500 = 0 \]
Factoring gives:
\[ (X - 25)(X + 20) = 0 \]

### Step 10: Solve for X
Setting each factor to zero gives us:
1. \( X - 25 = 0 \) → \( X = 25 \)
2. \( X + 20 = 0 \) → \( X = -20 \) (not valid since the number of students cannot be negative)

Thus, \( X = 25 \).

### Step 11: Calculate the Number of Students Who Attended
The number of students who attended the picnic is:
\[ X - 5 = 25 - 5 = 20 \]

### Step 12: Calculate the Cost Per Head for Attendees
The cost per head for the students who attended is:
\[ \text{Cost per head} = \frac{2000}{20} = 100 \text{ Rs.} \]

### Final Answer
- Number of students who attended the picnic: **20**
- Cost per head for each student: **Rs. 100**