Question

In a library, 50% of the total number of books are Marathi. The books of English are $\dfrac{1}{3}$rd of the Marathi books. The books on Mathematics are 25% of the English books. The remaining 560 books are of other subjects. What is the total number of books in the library?

Answer 1

Hint:This is a simple question of linear equations in one variable. We will assume the total number of books to be x. Now we will find the number of all the types of books in terms of x. Then we will add them and equate them to the total number of books in the library, that is x itself. We will then solve the equation to find x and get the answer.

Complete step-by-step answer:
We need to find the total number of books in the library. So, we will assume that the number of books in the library are x. We have been given that the number of Marathi books are 50% of the total books so-
Number of Marathi books = m = 50% of x = $\dfrac{{50}}{{100}} \times {\text{x}} = \dfrac{{\text{x}}}{2}$
Also, the number of english books are $\dfrac{1}{3}$rd of the Marathi books so-
Number of English books = e = $\dfrac{1}{3}{\text{m}} = \dfrac{1}{3} \times \dfrac{{\text{x}}}{2} = \dfrac{{\text{x}}}{6}$
Finally, the number of mathematics books are 25% of the number of English books so we can proceed as-
Number of mathematics books = M = 25% of e = $\dfrac{{25}}{{100}} \times {\text{e}} = \dfrac{1}{4} \times \dfrac{{\text{x}}}{6} = \dfrac{{\text{x}}}{{24}}$
Also, the remaining books in the library are 560, which is equal to r.

We can conclude that the total number of books are equal to the sum of Marathi, English, mathematics and the remaining books. Using this information, we can form an equation as-
x = m + e + M + r
Substituting these values, we get-
${\text{x}} = \dfrac{{\text{x}}}{2} + \dfrac{{\text{x}}}{6} + \dfrac{{\text{x}}}{{24}} + 560$
${\text{x}} - \dfrac{{\text{x}}}{2} - \dfrac{{\text{x}}}{6} - \dfrac{{\text{x}}}{{24}} = 560$
Taking 24 as the LCM, we get-
$\dfrac{{24{\text{x}} - 12{\text{x}} - 4{\text{x}} - {\text{x}}}}{{24}} = 560$
$\dfrac{{7{\text{x}}}}{{24}} = 560$
${\text{x}} = 560 \times \dfrac{{24}}{7} = 80 \times 24 = 1920$
These are the total number of books.

Note: The students often commit a mistake while assuming the number of mathematics or english books. They write their values such that they are $\dfrac{1}{3}$ of the total books, and not the marathi books. Similarly for mathematics books, they take the value as 25% of the total books, and not the english books.