some students planned for a picnic the budget for food was 500 rupee but 5 of them failed to go and thus the cost of food for each member increased by 5 rupee how many students attended the picnic

Let \( x \) be the original number of students who planned to attend the picnic.

1. The total budget for food is \( 500 \) rupees.
2. The initial cost per student would be \( \frac{500}{x} \) rupees.
3. However, since 5 students did not attend, the actual number of students attending is \( x - 5 \).
4. With fewer students, the cost per student increased by 5 rupees. So, the new cost per student is \( \frac{500}{x - 5} \).

According to the problem:
\[
\frac{500}{x - 5} = \frac{500}{x} + 5
\]

### Step 1: Set up the equation
\[
\frac{500}{x - 5} - \frac{500}{x} = 5
\]

### Step 2: Simplify the equation
To simplify, first find a common denominator:
\[
\frac{500x - 500(x - 5)}{x(x - 5)} = 5
\]
Expanding in the numerator:
\[
\frac{500x - 500x + 2500}{x(x - 5)} = 5
\]
This simplifies to:
\[
\frac{2500}{x(x - 5)} = 5
\]

### Step 3: Cross-multiply
\[
2500 = 5x(x - 5)
\]
\[
2500 = 5x^2 - 25x
\]
Divide both sides by 5:
\[
500 = x^2 - 5x
\]
Rearrange to form a quadratic equation:
\[
x^2 - 5x - 500 = 0
\]

### Step 4: Solve the quadratic equation
Using the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-500)}}{2 \cdot 1}
\]
\[
x = \frac{5 \pm \sqrt{25 + 2000}}{2}
\]
\[
x = \frac{5 \pm \sqrt{2025}}{2}
\]
Since \( \sqrt{2025} = 45 \):
\[
x = \frac{5 \pm 45}{2}
\]
So,
\[
x = \frac{50}{2} = 25 \quad \text{or} \quad x = \frac{-40}{2} = -20
\]

Since the number of students cannot be negative, \( x = 25 \).

### Conclusion
Originally, there were 25 students planned to attend the picnic. Since 5 students failed to attend, the number of students who actually attended is:
\[
25 - 5 = 20
\]

**Answer:** 20 students attended the picnic.

Would you like a breakdown of any part of this solution?

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**Relative Questions:**
1. What would happen if more students couldn’t attend, say 10?
2. How would the cost per student change if the budget was increased to 1000 rupees?
3. Can this problem be solved with a different method, like trial and error?
4. How would the solution change if only 3 students didn’t attend?
5. What if the increase in cost per student was 10 rupees instead of 5?

**Tip:** Quadratic equations often have two solutions, but only the positive solution typically makes sense in real-world scenarios involving counts of people.

Some students planned for a picnic. The budget for food was 500 rupees, but 5 of them failed to go, and thus the cost of food for each member increased by 5 rupees. How many students attended the picnic?

Explore how to determine the number of students who attended a picnic when the budget for food was 500 rupees, and the cost per student increased due to some not attending. This problem utilizes algebraic concepts and the quadratic formula. Suitable for Grades 7-9.

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