Let's solve this problem step-by-step:
1. Define Variables:
- Let the original number of students be [tex]\( n \)[/tex].
- The total budget for the food is Rs. 7200.
- The cost per student initially would be [tex]\( \frac{7200}{n} \)[/tex].
- When four students couldn't join, the number of students became [tex]\( n - 4 \)[/tex].
- The new cost per student after the reduction in number of students increased by Rs. 60, so it became [tex]\( \frac{7200}{n - 4} \)[/tex].
2. Create an equation reflecting the change in the cost per student:
- According to the problem, the new cost per student is the previous cost plus Rs. 60.
[tex]\[
\frac{7200}{n - 4} = \frac{7200}{n} + 60
\][/tex]
3. Solve the equation for [tex]\( n \)[/tex]:
- Rearrange the equation to:
[tex]\[
\frac{7200}{n - 4} - \frac{7200}{n} = 60
\][/tex]
- Find a common denominator and combine the terms on the left-hand side:
[tex]\[
\frac{7200n - 7200(n - 4)}{n(n - 4)} = 60
\][/tex]
- Simplify the numerator:
[tex]\[
\frac{7200n - 7200n + 28800}{n(n - 4)} = 60
\][/tex]
[tex]\[
\frac{28800}{n(n - 4)} = 60
\][/tex]
- Multiply both sides by [tex]\( n(n - 4) \)[/tex] to get rid of the denominator:
[tex]\[
28800 = 60n(n - 4)
\][/tex]
- Simplify:
[tex]\[
28800 = 60(n^2 - 4n)
\][/tex]
[tex]\[
480 = n^2 - 4n
\][/tex]
[tex]\[
n^2 - 4n - 480 = 0
\][/tex]
4. Solve the quadratic equation:
- Use the quadratic formula [tex]\( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -480 \)[/tex]:
[tex]\[
n = \frac{4 \pm \sqrt{16 + 1920}}{2}
\][/tex]
[tex]\[
n = \frac{4 \pm \sqrt{1936}}{2}
\][/tex]
[tex]\[
n = \frac{4 \pm 44}{2}
\][/tex]
- This gives us two solutions:
[tex]\[
n = \frac{48}{2} = 24 \quad \text{(positive solution)}
\][/tex]
[tex]\[
n = \frac{-40}{2} = -20 \quad \text{(negative, so we discard this)}
\][/tex]
5. Determine the number of students who went for the picnic:
- The original number of students is 24.
- Since 4 students couldn't join, the number of students who went for the picnic is:
[tex]\[
24 - 4 = 20
\][/tex]
Conclusion:
The original number of students planned for the picnic was 24, but since 4 students couldn’t join, the number of students who went for the picnic is 20.
