Question

Mr. Ram is on tour and he has Rs.360 for his expenses. If he exceeds his tour by 4 days, he must cut down daily expenses by ${\text{Rs}}3$. The number of days of Mr. Ram’s tour programme is:
A) 28 days
B) 24 days
C) 22 days
D) 20 days

Answer 1

Hint:
Here, we will first find Ram’s expense for each particular day by the unitary method. Then, we will form a mathematical equation showing the relationship between the original daily expense and the new daily expense after the cut down of daily expenses by ${\text{Rs}}3$. By solving this equation, we will be able to find the required number of days of Mr. Ram’s tour programme.

Complete step by step solution:
Let the number of days of Mr. Ram’s tour be $x$ days.
Mr. Ram is on tour and he has ${\text{Rs}}360$ for his expenses.
Therefore, his expenses for each day $ = {\text{Rs}}\dfrac{{360}}{x}$
Now, if he exceeds his tour by 4 days then, the number of days will become $\left( {x + 4} \right)$days.
So, his expenses for each day $ = {\text{Rs}}\dfrac{{360}}{{x + 4}}$
But, according to the question, in this case, he must cut down daily expenses by ${\text{Rs}}3$
Hence, this can be written mathematically as:
$\dfrac{{360}}{{x + 4}} = \dfrac{{360}}{x} - 3$
This equation clearly represents the situation that if he exceeds his tour by 4 days, he must cut down daily expenses by ${\text{Rs}}3$ where actually, his daily expenses were ${\text{Rs}}\dfrac{{360}}{x}$
Now, solving further, we get,
$ \Rightarrow \dfrac{{360}}{x} - \dfrac{{360}}{{x + 4}} = 3$
Now, taking LCM on the LHS and taking 360 common from the numerator, we get,
$ \Rightarrow 360\left( {\dfrac{{x + 4 - x}}{{x\left( {x + 4} \right)}}} \right) = 3$
Dividing both sides by 360, we get
$ \Rightarrow \left( {\dfrac{4}{{x\left( {x + 4} \right)}}} \right) = \dfrac{1}{{120}}$
Cross multiplying and solving further, we get,
$ \Rightarrow {x^2} + 4x = 480$
$ \Rightarrow {x^2} + 4x - 480 = 0$
Doing middle term split in this quadratic equation, we get
$ \Rightarrow {x^2} + 24x - 20x - 480 = 0$
$ \Rightarrow x\left( {x + 24} \right) - 20\left( {x + 24} \right) = 0$
Factoring out common terms, we get
$ \Rightarrow \left( {x - 20} \right)\left( {x + 24} \right) = 0$
By zero product property, we get
$ \Rightarrow x - 20 = 0$
$ \Rightarrow x = 20$
Or
 $ \Rightarrow x + 24 = 0$
$ \Rightarrow x = - 24$
But, the number of days can’t be negative. Therefore, rejecting the negative value, we get the number of days of Mr. Ram’s tour as 20 days.

Hence, option D is the correct answer.

Note:
To answer this question we should know how to represent given information, mathematically. Also, understanding the meaning of the question plays an important role in such types of questions because if for example, we understood the meaning incorrectly then we could write the equation as $\dfrac{{360}}{{x + 4}} - 3 = \dfrac{{360}}{x}$, hence, making our answer completely wrong.
We have used the zero product property which states that if the product of two terms is zero then either of them is zero.