Question

A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs. 22 for a book kept for six days, while Anand paid Rs. 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.
A) Rs. 13, Rs. 2
B) Rs. 10, Rs. 3
C) Rs. 12, Rs. 9
D) Rs. 14, Rs. 5

Answer 1

Hint: Here, we will assume ‘$x$’ = a fixed charge for the first two days. And, ‘$y$’ = charge for each extra day and then, transform the question into an equation by using these two variables. Latika paid Rs 22 for six days. We charge Rs. $x$ for the first two days and Rs. 4$y$ for the remaining 4 days i.e. ($x + 4y = 22$) and Anand paid Rs 16 for four days. We charge Rs. $x$ for the first two days and Rs. 2$y$ for the remaining 2 days i.e. ($x + 2y = 16$). Now solve these two equations to find out the value of $x$ & $y$.

Complete step by step solution:
Let, the fixed charge = Rs $x$\[\]
Charge for each extra day = Rs $y$
Case of Latika, equation will be
 $x + 4y = 22$
$ \Rightarrow $ $x = 22 - 4y$ …………... (1)
Because out of 6 days, the charge for the first two days are fixed, so we take ‘$x$’ for that and remaining for 4 days there is an extra charge so we take 4$y$ (For 1 day, Charge is Rs $y$. For 4 days, the charges will be 4$y$).
Case of Anand, equation will be
$x + 2y = 16$ ………... (2)
The reason is: For two days, the charge is fixed that is Rs. $x$. For the remaining 2 days, the charges will be 2$y$ (For 1 day, Charge is Rs $y$. For 2 days, the charges will be 2$y$).
Put the value of (1) in (2), we get
$22 - 4y + 2y = 16$
$ \Rightarrow $ $22 - 2y = 16$
$ \Rightarrow $ $22 - 16 = 2y$
$ \Rightarrow $ $6 = 2y$
$ \Rightarrow $ $y = \dfrac{6}{2} = 3$
Put the value of $y$ in (1)
$ \Rightarrow $ $x = 22 - 4\left( 3 \right)$
$ \Rightarrow $ $x = 10$

So, the fixed charge for the first two days is $Rs.10$, and charge for each extra day is $Rs.3 $.

Note:

Alternative approach:

Here, we can assume Rs. $x$ as a fixed charge for one day. So, charges for the first two days will be Rs. 2$x$. And, an additional charge for each day = Rs. $y$. Now, the equation is as follows based on the question.

For Latika

$2x + 4y = 22$

$2x = 22 - 4y$

$x = \dfrac{{22 - 4y}}{2}$ ……...  (1)

For Anand

$2x + 2y = 16$  ……...   (2)

Put the value of (1) in (2)

$2(\dfrac{{22 - 4y}}{2}) + 2y = 16$  

Here, by dividing 2 by 2, we get

$22 - 4y + 2y = 16$

$ \Rightarrow $ $22 - 2y = 16$

$ \Rightarrow $ $22 - 16 = 2y$

$ \Rightarrow $ $6 = 2y$

$ \Rightarrow $ $y = \dfrac{6}{2} = 3$

Put the value of $y$ in (1)

$ \Rightarrow $ $x = 22 - 4\left( 3 \right)$

$ \Rightarrow $ $x = 10$

So, the fixed charge for the first two days is Rs 10, for one day, the charge will be Rs. 5 ($\dfrac{{10}}{2}$) and charge for each extra day is Rs 3.