Question

Anirudh bought some books for Rs.60. If he had bought 5 more books for the same amount each book would have cost him 1 rupee less. Find the number of books bought by Anirudh and the price of each book.

Answer 1

Hint: In this question, we will proceed by considering the number of books bought by Anirudh to be \[x\]. Then we will find out the cost of each book he had already bought and the cost of each book if he had bought it. Then equate their difference to 1 rupee to get the required answer. So, use this concept to reach the solution to the given problem.

Complete step-by-step solution:
Let the number of books Anirudh bought = \[x\]
Given that the total cost of the books = Rs. 60
So, cost of each book = \[{\text{Rs}}{\text{.}}\dfrac{{60}}{x}\]
Also given that if Anirudh had bought 5 more books i.e., \[x + 5\] books the cost would be the same i.e., Rs. 60.
So, cost of each book if he had bought = \[{\text{Rs}}{\text{.}}\dfrac{{60}}{{x + 5}}\]
In the question, given that
The cost of each book he had already bought – the cost of each book if he had bought = Rs. 1
So, we have
\[ \Rightarrow \dfrac{{60}}{x} - \dfrac{{60}}{{x + 5}} = 1 \\
   \Rightarrow \dfrac{{60\left( {x + 5} \right) - 60\left( x \right)}}{{x\left( {x + 5} \right)}} = 1 \\
   \Rightarrow 60x + 60\left( 5 \right) - 60x = x\left( {x + 5} \right) \\
   \Rightarrow 300 = {x^2} + 5x \\
   \Rightarrow {x^2} + 5x - 300 = 0 \]
Splitting and taking the terms common, we have
\[\Rightarrow {x^2} + 20x - 15x - 300 = 0 \\
   \Rightarrow x\left( {x + 20} \right) - 15\left( {x + 20} \right) = 0 \\
   \Rightarrow \left( {x - 15} \right)\left( {x + 20} \right) = 0 \\
   \Rightarrow x = 15{\text{ or }} - 20 \]
Since the number of books can`t be a negative value, we have \[x = 15\].
Therefore, the cost of each book he had already bought \[ = \dfrac{{60}}{x} = \dfrac{{60}}{{15}} = {\text{Rs}}{\text{.4}}\]

Thus, the Anirudh had bought 15 books, and each cost 4 rupees.

Note: Observe the cost of each book if he had to be bought is less than the cost of the book he had already bought as the number of books if he had to be bought is 5 more than the books he had already bought for the same price. To check our answer, multiply the number of books bought by Anirudh with the cost of each book. If it is equal to the total cost of the books he had already bought i.e., Rs. 60 then our answer is correct otherwise incorrect.