Question

The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of $10$ km, the charge paid is $Rs.105$ and for a journey of $15$ km, the charge paid is $Rs.155$. What are the fixed charges and the charge per kilometer? How much does a person have to pay for traveling a distance of $25$ km?
$A)$ Fixed charge is $Rs.5$ and the charge per kilometer is $Rs.10$ for $25$km person has to pay $255$.
$B)$ Fixed charge is $Rs.10$ and the charge per kilometer is $Rs.5$ for $25$km person has to pay $135$.
$C)$ Fixed charge is $Rs.15$ and the charge per kilometer is $Rs.5$ for $25$km person has to pay $140$.
$D)$ Fixed charge is $Rs.50$ and the charge per kilometer is $Rs.20$ for $25$km person has to pay $50$.

Answer 1

In this question we have been provided with the data that the taxi charges consist of a fixed charge and the cost for the distance covered. We will consider the fixed cost to be $x$ and the cost for the distance covered to be $y$ per kilometer. We will then make two equations based on the two cases given to us and then solve the equations as a set of simultaneous equations and get the values. We will then calculate the value for a journey of $25$km.

Complete step by step solution:
Let the fixed charge and the cost for the distance covered be $x$ and $y$ respectively.
Now For a distance of $10$ km, the charge paid is $Rs.105$ therefore, we can write:
$\Rightarrow x+10y=105\to \left( 1 \right)$
And for a journey of $15$ km, the charge paid is $Rs.155$.
$\Rightarrow x+15y=155\to \left( 2 \right)$
On subtracting equation $\left( 1 \right)$ from $\left( 2 \right)$, we get:
$\Rightarrow 5y=50$
On dividing both the sides by $5$, we get:
$\Rightarrow y=10$, which is the cost per kilometer.
On substituting $y=10$ in equation $\left( 1 \right)$, we get:
$\Rightarrow x+10\left( 10 \right)=105$
On multiplying, we get:
$\Rightarrow x+100=105$
On transferring $100$ to the right-hand side, we get:
$\Rightarrow x=105-100$
On simplifying, we get:
$\Rightarrow x=5$, which is the fixed charge.
Now to find the cost for a distance of $25$km, we will multiply $25$ with the charge per kilometer and then add the fixed charge. Mathematically, we can write it as:
$\Rightarrow 25\times 10+5$
On simplifying, we get:
$\Rightarrow 255$, which is the required cost to travel $25$ kms therefore, the correct option is $\left( \text{A} \right)$.

Note: It is to be remembered that in any given equation multiplying or dividing the equation by a specific constant doesn’t change the value of the equation.
In the given question we had two variables which are $x$ and $y$, therefore they can be solved by using elimination, where there are more than three variables, and the matrix is used to solve them.