Question

A bus left with some definite no. of passengers. At the first stop half the passengers left the bus and 35 boarded the bus. At the second stop, one-fifth of the passengers left and 40 boarded the bus. Then the bus moved with 80 passengers towards its destination without stopping anywhere. How many passengers were there originally?

Answer 1

Hint: Here, we have to first assign some variables to the total number of passengers after every stop and for the number of passengers originally. Next step would be creating equations according to the conditions mentioned and solving them to find the required result.

Complete step by step answer:
Let the number of passengers in the bus be $x$, let the number of passengers in the bus after the first stop be ${{x}_{1}}$ and let the number of passenger in the bus after the second bus stop be ${{x}_{2}}$.
At the first stop, half the passengers left the bus and 35 other passengers boarded the bus which gives us
${{x}_{1}}=\dfrac{x}{2}+35$................(1)
Now, at the second stop, one-fifth of the passengers left the bus and 40 other passengers boarded the bus, we get
$\begin{align}
  & {{x}_{2}}={{x}_{1}}-\dfrac{{{x}_{1}}}{5}+40 \\
 & =\dfrac{5{{x}_{1}}-{{x}_{1}}}{5}+40 \\
 & =\dfrac{4{{x}_{1}}}{5}+40
\end{align}$
Finally, the bus had 80 passengers inside the bus and moved towards the destination without stopping anywhere and reached its destination.
The last stop was the second stop where the bus halted and where some passengers got down and some boarded the bus, and then the bus had 80 passengers after the second stop.
$\begin{align}
  & 80=\dfrac{4{{x}_{1}}}{5}+40 \\
 & 80=\dfrac{4{{x}_{1}}+200}{5} \\
 & 80\times 5=4{{x}_{1}}+200 \\
 & 400=4{{x}_{1}}+200
\end{align}$
Now, subtract with 200 on both the sides of the equation, we get
$\begin{align}
  & 400-200=4{{x}_{1}}+200-200 \\
 & 200=4{{x}_{1}}
\end{align}$
Now, divide by 4 on both sides of the equation, we will get the value of ${{x}_{1}}$.
$\begin{align}
  & \dfrac{200}{4}=\dfrac{4{{x}_{1}}}{4} \\
 & 50={{x}_{1}}
\end{align}$
Therefore, the total number of passengers in the bus after the first bus stop is 50 passengers.
Now, substitute the obtained value ${{x}_{1}}=50$, in equation (1) to find the total number of passengers originally boarded the bus.
$\begin{align}
  & 50=\dfrac{x}{2}+35 \\
 & 50=\dfrac{x+35\times 2}{2} \\
 & 50\times 2=x+70 \\
 & 100=x+70 \\
\end{align}$
Now, subtract by 70 on both the sides of the equation, we get
$\begin{align}
  & 100-70=x+70-70 \\
 & x=30 \\
\end{align}$

Hence, total number of passengers who were originally on the bus is 30.

Note: The above question is in the form of a linear equation with two variables. The general form of a linear equation with two variables is $ax+by=c$, where a, b and c are coefficients, $x$ and $y$ are the two variables.