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प्रश्न
| Some students planned a picnic. The total budget for hiring a bus was ₹1440. Later on, eight of them refused to go and instead paid their total share of money towards the fee of one economically weaker student of their class and thus, the cost for each member who went for picnic is increased by ₹30. |
1. If $$x$$ students planned for the picnic, then the share for hiring the bus per student who went for the picnic, was :
(a) ₹30x
(b) ₹1440x
(c) $$₹\frac{1440}{x}$$
(d) $$₹\left(\frac{1440}{x - 8}\right)$$
2. The algebraic representation of the given information in the form of a quadratic equation is:
(a) x2 – 8x – 384 = 0
(b) x2 + 8x – 384 = 0
(c) x2 – 8x – 184 = 0
(d) x2 + 8x – 184 = 0
3. How many students went for the picnic?
(a) 24
(b) 16
(c) 32
(d) 2
4. How much money was paid towards the fee?
(a) ₹280
(b) ₹340
(c) ₹420
(d) ₹ 480
5. What would be the share of each student if all the students had attended the picnic?
(a) ₹90
(b) ₹30
(c) ₹60
(d) none of these
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उत्तर
1. The correct option is (d).
Reasoning:
Total number of students who planned the picnic = x
Number of students who refused to go = 8
Number of students who actually went for the picnic = (x – 8)
Total cost for hiring the bus = ₹1440
Therefore, the share per student who went for the picnic is given by dividing the total cost by the number of students who attended: \[\text{Share per student} = ₹\left(\frac{1440}{x - 8}\right)\]
Option (c) gives the original planned share per student, not the share of those who actually went.
Options (a) and (b) incorrectly multiply the costs instead of dividing by the number of students.
2. The correct option is (a).
Reasoning:
Original planned share per student = \[\frac{1440}{x}\]
New share per attending student = \[\frac{1440}{x - 8}\]
According to the question, the share increases by ₹30:
\[\frac{1440}{x - 8} - \frac{1440}{x} = 30\]
\[1440 \left[\frac{1}{x - 8} - \frac{1}{x}\right] = 30\]
\[\frac{x - (x - 8)}{x(x - 8)} = \frac{30}{1440}\]
\[\frac{8}{x^2 - 8x} = \frac{1}{48}\]
\[x^2 - 8x = 8 \times 48\]
\[x^2 - 8x = 384\]
\[x^2 - 8x - 384 = 0\]
Thus, the quadratic equation representing the situation is \[x^2 - 8x - 384 = 0\].
3. The correct option is (b).
Reasoning: Solving the quadratic equation:
\[x^2 - 8x - 384 = 0\]
\[x^2 - 24x + 16x - 384 = 0\]
\[x(x - 24) + 16(x - 24) = 0\]
\[(x - 24)(x + 16) = 0\]
\[x = 24 \quad \text{or} \quad x = -16\]
Since the number of students cannot be negative, x = 24.
Total students who originally planned = 24
Number of students who went for the picnic = x – 8 = 24 – 8 = 16
Thus, 16 students went for the picnic.
4. The correct option is (d).
Reasoning:
Original planned share per student = \[\frac{1440}{x} = \frac{1440}{24} = ₹60\]
Eight students refused to go and instead paid their total share of money towards the fee.
Total money paid towards the fee = \[8\ \times ₹60 = ₹480\]
Alternatively, this is equal to the increase in total cost borne by the remaining 16 students: \[16\ \times ₹30 = ₹480\].
5. The correct option is (c).
Reasoning:
Total budget = ₹1440
Total number of students who planned the picnic = x = 24
If all 24 students had attended, the share of each student would be: \[\text{Share} = \frac{1440}{24} = ₹60\]
