Simple Interest Rate Calculation

The problem asks for the annual simple interest rate (R) when an initial amount (Principal, P) grows to a new amount (A) in a given time (T).

Identifying Key Information

• Let the Principal amount be P.
• The final Amount (A) becomes eight-fifth of the original: $A = \frac{8}{5}P$.
• The Time period (T) is 5 years.
• We need to find the Rate (R) of simple interest per annum.

Calculating Simple Interest (SI)

The Simple Interest earned is the difference between the final Amount and the Principal:

$ SI = A - P $

Substituting the value of A:

$ SI = \frac{8}{5}P - P $

To subtract, find a common denominator:

$ SI = \frac{8}{5}P - \frac{5}{5}P $

$ SI = \frac{8 - 5}{5}P $

$ SI = \frac{3}{5}P $

Applying the Simple Interest Formula

The formula for Simple Interest is:

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

We know SI = $\frac{3}{5}P$ and T = 5 years. Substitute these values into the formula:

$ \frac{3}{5}P = \frac{P \times R \times 5}{100} $

Solving for the Rate (R)

We can cancel out P from both sides of the equation (assuming P is not zero):

$ \frac{3}{5} = \frac{R \times 5}{100} $

Now, solve for R. First, simplify the right side:

$ \frac{3}{5} = \frac{5R}{100} $

Multiply both sides by 100 to isolate 5R:

$ \frac{3}{5} \times 100 = 5R $

$ 3 \times 20 = 5R $

$ 60 = 5R $

Divide by 5 to find R:

$ R = \frac{60}{5} $

$ R = 12 $

Therefore, the simple interest rate is 12% per annum.