Calculating Simple Interest Rate When Principal Doubles

This question asks us to find the annual simple interest rate when a fixed sum of money (principal) doubles itself in 13 years. Understanding the concepts of simple interest, principal, amount, and time is crucial here.

Let's define the terms:

• Principal (P): The initial sum of money.
• Amount (A): The total sum of money after a certain time, including the principal and the interest earned.
• Interest (I): The extra money earned on the principal over time.
• Time (T): The duration for which the money is invested or borrowed, usually in years.
• Rate (R): The annual interest rate, usually expressed as a percentage.

The relationship between Amount, Principal, and Interest is:

\(A = P + I\)

The formula for calculating simple interest is:

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

In this problem, we are given that the fixed sum (Principal, P) doubles in 13 years. This means the Amount (A) after 13 years is double the Principal. So,

\(A = 2P\)

Now, we can find the interest earned (I) by substituting \(A = 2P\) into the formula \(A = P + I\):

\(2P = P + I\)

Subtracting P from both sides, we get:

\(I = 2P - P\)

\(I = P\)

So, the interest earned is equal to the original principal amount.

Now we can use the simple interest formula \(I = \frac{P \times R \times T}{100}\) and substitute the values we know: \(I = P\) and \(T = 13\) years.

\(P = \frac{P \times R \times 13}{100}\)

We need to solve for R. Assuming the principal P is not zero, we can divide both sides of the equation by P:

\(1 = \frac{R \times 13}{100}\)

Now, multiply both sides by 100 to isolate \(R \times 13\):

\(1 \times 100 = R \times 13\)

\(100 = 13R\)

Finally, divide both sides by 13 to find the rate R:

\(R = \frac{100}{13}\)

Let's calculate the value of R:

\(R = 7.692307...\)

The question asks for the interest rate per year, correct to two decimal places. Rounding 7.692307... to two decimal places gives 7.69.

Therefore, the interest rate per year is approximately 7.69%.

This calculation confirms that if a sum doubles at simple interest in 13 years, the annual interest rate is 7.69% when rounded to two decimal places.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Doubling Time

A common question in simple interest involves the time it takes for the principal to double, triple, or become 'n' times itself. For simple interest, if a principal doubles (A=2P), the interest earned is I=P. Using \(I = \frac{PRT}{100}\), we get \(P = \frac{PRT}{100}\), which simplifies to \(1 = \frac{RT}{100}\) or \(RT = 100\). This gives a simple relationship:

• If you know the Rate (R), the Time (T) to double is \(T = \frac{100}{R}\) years.
• If you know the Time (T), the Rate (R) required to double is \(R = \frac{100}{T}\) % per year.

In our specific problem, we were given T = 13 years, so the rate R is indeed \(R = \frac{100}{13}\) %.

If the principal were to triple (A=3P), then I = 2P. The formula becomes \(2P = \frac{PRT}{100}\), simplifying to \(2 = \frac{RT}{100}\) or \(RT = 200\). In general, if the amount becomes 'n' times the principal (A=nP), the interest is \((n-1)P\), leading to the relationship \(RT = (n-1) \times 100\).