Question

A person on tour has $Rs.360$ for his expenses. If he extends his tour for $4$ days, he has to cut down his daily expenses by $Rs.3$. Find the original duration of the tour.

Answer 1

Hint: Find the expenditure per day for both the initial tour and the tour after extension and then satisfy the condition given in the question.

Complete step-by-step answer:
Let the original duration of the tour be $x$ days.
Given, total expenditure of the tour $ = Rs.360$
Expenditure per day $ = Rs.\dfrac{{360}}{x}$
Now, the person has to extend his duration by $4$ days.
Duration of the extended tour $ = \left( {x + 4} \right)$ days.
Expenditure per day according to the new schedule $ = Rs.\dfrac{{360}}{{x + 4}}$
In the question, it is given that the daily expenses are cut down by $Rs.3$. Therefore we have:
$
   \Rightarrow \dfrac{{360}}{x} - \dfrac{{360}}{{x + 4}} = 3, \\
   \Rightarrow \dfrac{{360x + 1440 - 360x}}{{x\left( {x + 4} \right)}} = 3, \\
   \Rightarrow 3x\left( {x + 4} \right) = 1440, \\
   \Rightarrow {x^2} + 4x = 480, \\
   \Rightarrow {x^2} + 4x - 480 = 0 \\
$
Thus, the above equation is a quadratic equation in $x$. We’ll get $2$ values of$x$:
$
   \Rightarrow {x^2} + 4x - 480 = 0, \\
   \Rightarrow {x^2} + 24x - 20x - 480 = 0, \\
   \Rightarrow x\left( {x + 24} \right) - 20\left( {x + 24} \right) = 0, \\
   \Rightarrow \left( {x - 20} \right)\left( {x + 24} \right) = 0 \\
$
Thus we have $x = 20$ or $x = - 24$.
But the number of days can never be negative. So, $x = 20$ is the only plausible solution.
Therefore, the original duration of the tour is $20$ days.

Note: We can solve the above question in an alternative way as:
Let $x$ be the initial number of days of tour. Then after extension, the number of days is \[\left( {x + 4} \right)\].
Further, let $r$ be his initial daily expenses. Then after the extension of the tour, daily expenses will be reduced by $3$. So, it will be $\left( {r - 3} \right)$.
But the total expense remains constant and it is $360$. So, based on these information we can form equations:
$ \Rightarrow x \times r = \left( {x + 4} \right) \times \left( {r - 3} \right) = 360$
From this we will get two equations. On solving, it will give the same result.