Question

A person on tour has ₹ 5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹ 180. Find the original duration of the tour and daily expense.

Hint:

For word problems leading to quadratics, often identifying the "difference" equation (e.g., \(E_1 - E_2 = \text{diff}\)) is the easiest way to set up the problem correctly.

Answer 1

Step 1: Understanding the Situation:
Total budget = ₹5400 (fixed).
When the duration increases by 5 days, the daily expense decreases by ₹180.

Step 2: Let the Original Duration be:
Let original number of days = n

Original daily expense = 5400 / n
New duration = n + 5
New daily expense = 5400 / (n + 5)

Given:
Original expense − New expense = 180

5400/n − 5400/(n + 5) = 180

Step 3: Solving the Equation:
Divide the entire equation by 180:
30/n − 30/(n + 5) = 1

Take LCM:
30[(n + 5 − n) / n(n + 5)] = 1
30(5) / [n(n + 5)] = 1
150 / (n² + 5n) = 1

So,
n² + 5n = 150
n² + 5n − 150 = 0

Factorising:
n² + 15n − 10n − 150 = 0
n(n + 15) − 10(n + 15) = 0
(n − 10)(n + 15) = 0

n = 10 or n = −15

Since duration cannot be negative,
n = 10

Step 4: Finding Daily Expense:
Daily expense = 5400 / 10
= ₹540

Final Answer:
Original duration = 10 days
Original daily expense = ₹540