A certain sum amounts to Rs.4500 in \[\left( 2\dfrac{1}{2} \right)year\] and 20% per annum simple interest. Find the sum (in Rs.)

Hint: In the above question, we will first find the simple interest and add it to the principal value which is equal to the sum amount and by equating it to the given numerical value of the sum amount we will find the value of sum. The formula of simple interest is as follows:
\[\text{S}\text{.I = }\dfrac{P\times r\times t}{100}\]
Where, P is principal amount, r is rate and t is time.
Also, we will use the formula for sum amount given by,
\[\text{Sum = P + S}\text{.I}\]
Where, P is principal amount and I is simple interest.

Complete step-by-step answer:
We have been given that, a certain sum amounts to Rs.4500 in \[\left( 2\dfrac{1}{2} \right)year\] at 20% per annum simple interest, so we have to find the sum i.e. principal value
We know that, simple interest \[SI\text{ = }\dfrac{P\times r\times t}{100}\]
Let us take the principal value to be Rs.x. Now, we have all the necessary value, so we can substitute in the above equation as shown below,
\[\begin{align}
  & S.I=\dfrac{x\times 20\times \left( 2\dfrac{1}{2} \right)}{100} \\
 & \Rightarrow \dfrac{x\times 20\times \dfrac{5}{2}}{100} \\
 & \Rightarrow \dfrac{x}{2} \\
\end{align}\]
Now, we know that, sum amount = Principal value + Simple interest
\[\Rightarrow \text{Sum amount = x+}\dfrac{x}{2}\]
On taking LCM of the terms, we get:
\[\begin{align}
  & 4500=x+\dfrac{x}{2} \\
 & 4500=\dfrac{2x+x}{2}=\dfrac{3x}{2} \\
\end{align}\]
On cross multiplying the equation by \[\dfrac{2}{3}\] we get:
\[\begin{align}
  & \dfrac{2}{3}\times 4500=x \\
 & 3000=x \\
\end{align}\]
Therefore, the sum is equal to Rs.3000

Note: Be careful while solving the question because you may treat the sum amount as the principal value and you move further to find the sum of the amount but it is incorrect as we have been given the sum amount and asked to find the value of principal amount.