Question

From a bus stand in Bangalore, if we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur, the total cost is Rs. 46; but if we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur the total cost is Rs. 74. Find the fares from the bus stand to Malleswaram and to Yeshwanthpur.

Answer 1

Hint: Assume the ticket price to Malleswaram be x and to Yeshwanthpur be y. Now calculate the total price in terms of x, y. In case first, we have 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur which costs Rs. 46. So, the total price is \[2x+3y=46\] . In case second, we have 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur which costs Rs. 74. So, the total price is \[3x+5y=74\]. Now, we have to find the values of x and y. We also have two equations. Solve these equations and get the values of x and y.

Complete step-by-step solution -
According to the question, we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur, the total cost is Rs. 46 and if we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur the total cost is Rs. 74. Analyze both cases separately.
Let us assume the ticket price from Bangalore to Malleswaram be and from Bangalore to Yeshwanthpur be y.
In the case 1st, we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur
Cost of 2 tickets from Bangalore to Malleswaram = 2x
Cost of 3 tickets from Bangalore to Yeshwanthpur = 3y
The total cost of the tickets = \[2x+3y\] .
It is given that the total cost is Rs. 46.
So, \[2x+3y=46\] …………………(1)
Similarly, In the case 2nd, we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur
Cost of 3 tickets from Bangalore to Malleswaram = 3x
Cost of 5 tickets from Bangalore to Yeshwanthpur = 5y
The total cost of the tickets = \[3x+5y\] .
It is given that the total cost is Rs. 74.
So, \[3x+5y=74\] …………………(2)
Now we have two equations and we have to find the values of x and y.
From equation (1), find the value of x in terms of y.
\[\begin{align}
  & 2x+3y=46 \\
 & \Rightarrow 2x=46-3y \\
\end{align}\]
\[\Rightarrow x=\left( \dfrac{46-3y}{2} \right)\] …………………..(3)
Now, using equation (3), putting the value of x in equation (2), we get
\[3x+5y=74\]
\[\begin{align}
  & \Rightarrow 3\left( \dfrac{46-3y}{2} \right)+5y=74 \\
 & \Rightarrow 138-9y+10y=148 \\
\end{align}\]
\[\Rightarrow y=10\] ……………..(4)
From equation (4), putting the values of y in equation (1), we get
\[\begin{align}
  & 2x+3y=46 \\
 & \Rightarrow 2x+30=46 \\
 & \Rightarrow 2x=16 \\
\end{align}\]
\[\Rightarrow x=8\] ……………….(5)
From equation (4) and equation (5), we have got the values of x and y.
Therefore, the ticket price for Bangalore to Malleswaram is Rs. 8 and for Bangalore to Yeshwanthpur is Rs. 10.

Note: We can also solve this question in another way. Assume the ticket price to Malleswaram be x and to Yeshwanthpur be y. Now, we have two equations.
\[2x+3y=46\] ……………..(1)
\[3x+5y=74\] ………………(2)
Multiply by 3 in equation (1) and multiply by 2 in equation (2).
\[6x+9y=138\] …………………..(3)
\[6x+10y=148\] ……………………..(4)
Subtracting equation (4) from equation (3), we get
\[y=10\]……….(5)
Now, put the value of y in equation (1), we get
\[\begin{align}
  & 2x+3y=46 \\
 & \Rightarrow 2x+30=46 \\
 & \Rightarrow 2x=16 \\
\end{align}\]
\[\Rightarrow x=8\]
Hence, the ticket price for Bangalore to Malleswaram is Rs. 8 and for Bangalore to Yeshwanthpur is Rs. 10.