Tour-budget planning: Mr. Arjun has ₹ 360 for his trip. If he extends his tour by 4 days, he must reduce his daily expense by ₹ 3 to stay within budget. For how many days was the original plan?
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A40
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B20
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C60
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D15
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E24
Answer
Correct Answer: 20
Explanation
Introduction / Context: This is a classic unitary-method / algebra blend about fixed total budget and varying duration. Extending the number of days forces a reduction in daily spending so that the total cost remains the same. We form an equation in the number of days and solve it cleanly.
Given Data / Assumptions:
- Total budget B = ₹ 360.
- Original number of days = d; original daily expense e = 360/d.
- Extended trip: d + 4 days; new daily expense e − 3 = 360/(d + 4).
- d is a positive integer.
Concept / Approach: Equate the two expressions for the new daily expense. This yields a rational equation in d that simplifies to a quadratic. Solve for d and choose the positive, meaningful value.
Step-by-Step Solution:
Original daily expense e = 360/d.New daily expense e − 3 = 360/(d + 4).So 360/d − 3 = 360/(d + 4).Bring to one side: 360/d − 360/(d + 4) = 3.Compute LHS: 360[(d + 4) − d] / (d(d + 4)) = 1440 / (d(d + 4)).Thus 1440 / (d(d + 4)) = 3 ⇒ d(d + 4) = 480.d^2 + 4d − 480 = 0 ⇒ Discriminant = 16 + 1920 = 1936 = 44^2.d = (−4 + 44)/2 = 20 (take positive root).Verification / Alternative check: Original e = 360/20 = ₹ 18/day. New duration 24 days with ₹ 15/day gives total 24*15 = ₹ 360, consistent.
Why Other Options Are Wrong: 40, 60, 15, 24 do not satisfy the budget relation; only d = 20 preserves the total expense under the given change.
Common Pitfalls: Algebra slips when clearing denominators; forgetting that total budget is fixed in both scenarios.
Final Answer: 20