Question

Two shelves contain $55$ books. If half of the book from the second shelf were relocated to the first shelf, then the first shelf would contain $4$ times more books than the second one. How many books are there on each shelf $?$

Answer 1

Step-1: Forming equation using given condition:
Let $\mathrm{x}$ and $\mathrm{y}$ be the number of books in the first and second shelf respectively.

Since the total number of books in the first and second shelves is $55$, it follows that:

$$\begin{gathered} x+y=55 \\ \Rightarrow y=55-x\begin{array}{}\\ \end{array} \end{gathered}$$

So, the number of books in the second shelf are $55-\mathrm{x}$.

Half of books in the second shelf will be$\frac{55-\mathrm{x}}{2}$

After relocation of half of the books, the number of books left in the second shelf will be:

$\frac{55-\mathrm{x}}{2}$

Then, the number of book in the first shelf after relocation:

$\begin{array}{rcl}\mathrm{x}+\frac{55-\mathrm{x}}{2}& =& \frac{2\mathrm{x}+55-\mathrm{x}}{2}\\ & =& \frac{55+\mathrm{x}}{2}\end{array}$

Step-2: Find number of books in each shelf:

Since the first shelf have $4$ times more books than the second one, it follows that:

$$\begin{gathered} \frac{55+\mathrm{x}}{2}=4\times \left(\frac{55-\mathrm{x}}{2}\right) \\ \Rightarrow 55+\mathrm{x}=4\times (55-\mathrm{x}) \\ \Rightarrow 55+\mathrm{x}=220-4\mathrm{x} \\ \Rightarrow 5\mathrm{x}=165 \\ \Rightarrow x=33 \end{gathered}$$

Substitute the value of $\mathrm{x}$ in the first equation to get the value of $\mathrm{y}$:

$\begin{array}{rcl}\mathrm{y}& =& 55-33\\ & =& 22\end{array}$

Hence, the number of books in the first shelf is $\mathbf{33}$ and the number of books in the second shelf is $\mathbf{22}$.