Directions: Solve the question below and select the correct option.

The simple interest earned after 5 years on an investment of Rs (x + 4800) at an annual rate of 20% is equal to the total amount accumulated by investing Rs x at a compound interest rate of 40% per annum, compounded annually, for 2 years. Find the value of x.

The correct option is 5000.

Given Data:
Simple Interest Principal (P₁) = Rs (x + 4800)
Rate of Simple Interest (R₁) = 20% per annum
Time period for Simple Interest (T₁) = 5 years

Compound Interest Principal (P₂) = Rs x
Rate of Compound Interest (R₂) = 40% per annum (compounded annually)
Time period for Compound Interest (T₂) = 2 years

Step 1: Calculate the Simple Interest (SI) earned
The formula for calculating Simple Interest is:

$\text{SI}=\frac{{\text{P}}_1\times {\text{R}}_1\times {\text{T}}_1}{100}$

Substituting the values into the formula:
$\text{SI}=\frac{(x+4800)\times 20\times 5}{100}$

$\text{SI}=\frac{(x+4800)\times 100}{100}$

$\text{SI}=x+4800$

Step 2: Calculate the total accumulated amount (A) under Compound Interest
The formula for the total accumulated amount compounded annually is:

$\text{A}={\text{P}}_2{(1+\frac{{\text{R}}_2}{100})}^{{\text{T}}_2}$

Substituting the values into the formula:
$\text{A}=x{(1+\frac{40}{100})}^2$

$\text{A}=x{(1+0.4)}^2$

$\text{A}=x{\left(1.4\right)}^2$

$\text{A}=1.96x$

Step 3: Equate Simple Interest to the Compound Interest Amount
According to the problem statement, the Simple Interest earned is equal to the total amount accumulated under Compound Interest:

$\text{SI}=\text{A}$

$x+4800=1.96x$

Rearranging the equation to solve for x:
$1.96x-x=4800$

$0.96x=4800$

$x=\frac{4800}{0.96}$

$x=5000$

Therefore, the value of x is 5000.