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Question: Aniruddha bought some books for Rs. 60. Had he bought 5 more books for the same amount, each book wo...

Aniruddha bought some books for Rs. 60. Had he bought 5 more books for the same amount, each book would have cost him 1 rupee less. Find the number of books bought by Aniruddha.

Explanation

Solution

Hint: Assume that the number of books bought by Aniruddha equals ‘n’. He bought ‘n’ books for Rs 60. So, the price of each book is 60n\dfrac{60}{n}. It is given that if he had bought 5 more books i.e. ‘n+5’ books for the same price i.e. Rs 60, each book would have cost him Rs. 60n+5\dfrac{60}{n+5}.
Using this data and the information given in the question, solve for ‘n’.

Let us assume that Aniruddha has bought n books. In the question, it is given that he bought this book for Rs. 60. So, the price of each book will be equal to Rs 60n........(1)\dfrac{60}{n}........\left( 1 \right).
If he would have bought 5 more books i.e. n+5 books for the same price i.e. Rs 60, the price of each book will be equal to 60n+5...........(2)\dfrac{60}{n+5}...........\left( 2 \right)
Also, it is given in the question that if he would have 5 more books i.e. n+5 books for the same price i.e. Rs 60, each book would have cost him 1 rupee less.
Using equation (1)\left( 1 \right) and (2)\left( 2 \right) and the information mention in the above paragraph, we get,
60n−1=60n+5 ⇒60−nn=60n+5 ⇒(60−n)(n+5)=60n ⇒60n+300−n2−5n=60n ⇒n2+5n−300=0 ⇒n2−15n+20n−300=0 ⇒n(n−15)+20(n−15)=0 ⇒(n−15)(n+20)=0 ⇒n=15,n=−20 \begin{aligned} & \dfrac{60}{n}-1=\dfrac{60}{n+5} \\\ & \Rightarrow \dfrac{60-n}{n}=\dfrac{60}{n+5} \\\ & \Rightarrow \left( 60-n \right)\left( n+5 \right)=60n \\\ & \Rightarrow 60n+300-{{n}^{2}}-5n=60n \\\ & \Rightarrow {{n}^{2}}+5n-300=0 \\\ & \Rightarrow {{n}^{2}}-15n+20n-300=0 \\\ & \Rightarrow n\left( n-15 \right)+20\left( n-15 \right)=0 \\\ & \Rightarrow \left( n-15 \right)\left( n+20 \right)=0 \\\ & \Rightarrow n=15,n=-20 \\\ \end{aligned}
Since n represents the number of books, so, it must be positive.

⇒n=15\Rightarrow n=15
Hence, the answer is 15.

Note: There is a possibility that one may commit a mistake by writing the equation as
60n=60n+5−1\dfrac{60}{n}=\dfrac{60}{n+5}-1 instead of 60n−1=60n+5\dfrac{60}{n}-1=\dfrac{60}{n+5}. So, one should read this question very carefully in order to obtain the correct equation and the correct answer.