Question

The pair of 3 full tickets and two half tickets is Rs.204 and the pair of two full tickets and 3 half tickets is Rs.186. Find the cost of a pair of a full ticket and a half ticket.

Answer 1

Hint: In this question, we are given the price of 3 full tickets and two half tickets and the price of the pair of two full tickets and 3 half tickets. We can take the price of a full ticket to be x and the price of a half ticket to be y and then solve the resulting equations to find the value of x and y. As we have to find the cost of a pair of a full ticket and a half ticket, the answer will then be equal to the value of x+y.

Complete step-by-step answer:
Let the price of a full ticket be Rs.x and the price of a half ticket be Rs.y………………………(1.1)
It is given that the price of 3 full tickets and two half tickets is Rs.204. Thus, from (1.1), we can write this as
$3x+2y=204$
Multiplying both sides by 2 we obtain
$\begin{align}
  & 2\left( 3x+2y \right)=2\times 204 \\
 & 6x+4y=408......................................(1.2) \\
\end{align}$
Also, It is given that the price of 2 full tickets and 3 half tickets is Rs.186. Thus, from (1.1), we can write this as
$2x+3y=186$
Multiplying both sides by 3, we obtain
$\begin{align}
  & 3\left( 2x+3y \right)=3\times 186 \\
 & \Rightarrow 6x+9y=558.......................(1.3) \\
\end{align}$
Now, subtracting equation (1.2) from equation (1.3), we obtain
$\begin{align}
  & \left( 6x+9y \right)-\left( 6x+4y \right)=558-408 \\
 & \Rightarrow 5y=150 \\
 & \Rightarrow y=\dfrac{150}{5}=30........................(1.4) \\
\end{align}$
Using this value of y in equation (1.2), we obtain
$\begin{align}
  & 6x+4y=408 \\
 & \Rightarrow 6x+4\times 30=408 \\
 & \Rightarrow 6x=408-120=288 \\
 & \Rightarrow x=\dfrac{288}{6}=48.......................(1.5) \\
\end{align}$
Thus, from equation (1.4) and (1.5), we obtain the price of a full ticket to be Rs.x=Rs48 and that of a half ticket to be Rs.y=Rs.30. Thus,
Price of a pair of a full ticket and a half ticket= Price of a full ticket + Price of a half ticket
= Rs.x + Rs.y =Rs.48+ Rs.30= Rs.78
Thus, Rs. 78 is the required answer.

Note: We should note that we cannot take the price of a half ticket to be half of the price of a full ticket because commercially they are generally different. Also, we can solve the linear equations (1.2) and (1.3) by any known method such as substitution, elimination, cross multiplication etc. However, the answer would have remained the same.