Finding Simple Interest Rate When Sum Doubles

Let's break down this simple interest problem step-by-step. We are told that a sum of money doubles itself in 5 years due to simple interest. We need to find the annual rate of interest.

In simple interest problems, we deal with the following terms:

• Principal (P): The initial amount of money.
• Amount (A): The total money after adding the interest to the principal.
• Simple Interest (SI): The interest earned only on the principal amount over a period of time.
• Rate (R): The annual rate of interest (usually in percent).
• Time (T): The duration for which the money is invested or borrowed (usually in years).

According to the question, the sum becomes its double in 5 years. This means:

• The initial sum (Principal) is P.
• The final amount (Amount) is double the principal, so A = 2P.
• The time period (T) is 5 years.

The Simple Interest (SI) earned is the difference between the Amount and the Principal:

\( \text{SI} = \text{A} - \text{P} \)

Substituting the values we know (A = 2P):

\( \text{SI} = 2\text{P} - \text{P} \)

\( \text{SI} = \text{P} \)

So, the simple interest earned over 5 years is equal to the initial principal amount.

Using the Simple Interest Formula

The formula for calculating Simple Interest is:

\( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \)

We know SI = P, P = P, and T = 5 years. We need to find R.

Let's plug these values into the formula:

\( \text{P} = \frac{\text{P} \times \text{R} \times 5}{100} \)

Now, we need to solve this equation for R. We can cancel P from both sides of the equation (assuming P is not zero, which it must be for there to be interest).

\( 1 = \frac{\text{R} \times 5}{100} \)

To isolate R, we can multiply both sides by 100:

\( 1 \times 100 = \text{R} \times 5 \)

\( 100 = 5\text{R} \)

Now, divide both sides by 5:

\( \text{R} = \frac{100}{5} \)

\( \text{R} = 20 \)

The rate of interest is 20% per annum.

Let's verify this:

If P is the principal and R is 20% and T is 5 years, Simple Interest would be:

\( \text{SI} = \frac{\text{P} \times 20 \times 5}{100} = \frac{\text{P} \times 100}{100} = \text{P} \)

The Amount would be A = P + SI = P + P = 2P, which is double the principal. This matches the condition in the question.

Simple Interest Calculation Summary

Using \( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \), we found R = 20%.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest

Simple interest is a basic concept in finance. It's easier to calculate compared to compound interest because the principal amount remains constant throughout the loan or investment period for interest calculation.

• Many short-term loans use simple interest.
• Understanding simple interest is crucial before learning about compound interest, where interest is calculated on the principal amount plus the accumulated interest from previous periods.
• If the time is given in months, convert it to years by dividing by 12. If given in days, convert to years by dividing by 365 (or 366 for a leap year, though usually 365 is assumed unless specified).
• The formula \( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \) can be rearranged to find any of the variables if the others are known. For example, \( \text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}} \).

In this problem, the fact that the sum doubles (A=2P) directly tells us that the simple interest earned (SI = A - P) is equal to the principal (SI=P). This relationship simplifies finding the rate.