Question

At the first stop on his route, a driver unloaded $\dfrac{2}{5}$ of the packages in his van. After he unloaded another 3 packages, at his next stop $\dfrac{1}{2}$ of the original number of packages remained. How many packages were in the van before the first delivery?
A. 25
B. 10
C. 30
D. 36

Answer 1

Hint: We will start solving this question by assuming the total number of packages in the van to be $x$. Then by using the information given in the question that at the first stop driver unloaded $\dfrac{2}{5}$ of the packages and after he unloaded another 3 packages, at his next stop $\dfrac{1}{2}$ of the original number of packages remained, we form equations. By solving the equations obtained we get the desired answer.

Complete step-by-step solution:
We have been given that a driver unloaded $\dfrac{2}{5}$ of the packages in his van at his first stop. After he unloaded another 3 packages, at his next stop $\dfrac{1}{2}$ of the original number of packages remained.
We have to find the total number of packages in the van before the first delivery.
Let us assume that the total number of packages in the van before the first delivery is $x$.
Now, we have $\dfrac{1}{2}$ of the original number of packages remained after the driver unloaded $\dfrac{2}{5}$ of the packages in his van at his first stop and another 3 packages, at his next stop.
So, when we write mathematically we have
$\dfrac{1}{2}x=\dfrac{2}{5}x+3$
Now, solving the above expression we get
$\begin{align}
  & \Rightarrow \dfrac{x}{2}=\dfrac{2x}{5}+3 \\
 & \Rightarrow \dfrac{x}{2}-\dfrac{2x}{5}=3 \\
\end{align}$
Now, by taking LCM and solving further we get
$\begin{align}
  & \Rightarrow \dfrac{x\times 5-2x\times 2}{10}=3 \\
 & \Rightarrow \dfrac{5x-4x}{10}=3 \\
 & \Rightarrow \dfrac{x}{10}=3 \\
\end{align}$
Now, by cross multiplying we get
$\begin{align}
  & \Rightarrow x=3\times 10 \\
 & x=30 \\
\end{align}$
So, the total number of packages in the van before the first delivery is $30$.
Option C is the correct answer.

Note: The possibility of mistake in these types of questions is in the forming of equations. An incorrect equation leads to an incorrect solution. There is a possibility that students forget to multiply $\dfrac{2}{5}$ with x. We must note that $\dfrac{2}{5}$ of packages and 3 packages denote different things. The first represents a portion of the entire packages and the latter represents the number of packages. So read the information given carefully and form the equation.