Understanding Simple Interest and Doubling Money

This question asks us to find the rate of simple interest per annum at which a sum of money will double itself in a specific period, which is 12 years. Simple interest is calculated only on the initial principal amount.

Let's break down the problem using the key components of simple interest:

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

Calculating the Simple Interest Earned

We are told that the sum of money "doubles itself" in 12 years. This means that the final amount (A) is double the initial principal (P).

So, we have:

\(A = 2 \times P\)

The simple interest earned is the difference between the final amount and the principal:

\(SI = A - P\)

Substituting \(A = 2P\):

\(SI = 2P - P\)

\(SI = P\)

This shows that when money doubles under simple interest, the total simple interest earned is equal to the original principal amount.

Using the Simple Interest Formula to Find the Rate

The formula for calculating simple interest is:

\[SI = \frac{P \times R \times T}{100}\]

We know the following values:

• Simple Interest (SI) = \(P\)
• Principal (P) = \(P\) (We can use 'P' as the symbol for principal)
• Time (T) = 12 years
• Rate (R) = ? (This is what we need to find)

Now, let's substitute these values into the simple interest formula:

\[P = \frac{P \times R \times 12}{100}\]

To solve for R, we can rearrange the equation. Since P is on both sides and represents the principal (which is assumed to be a positive value), we can divide both sides by P:

\[1 = \frac{R \times 12}{100}\]

Now, multiply both sides by 100:

\[1 \times 100 = R \times 12\]

\[100 = 12R\]

Finally, divide both sides by 12 to find R:

\[R = \frac{100}{12}\]

Let's simplify the fraction \(\frac{100}{12}\). Both numbers are divisible by 4:

\[R = \frac{100 \div 4}{12 \div 4} = \frac{25}{3}\]

To express this as a decimal or percentage, we divide 25 by 3:

\[\frac{25}{3} \approx 8.333...\)

So, the rate per annum is approximately 8.33%.

Verifying the Rate with Options

The calculated rate is approximately 8.33%. Let's look at the provided options:

The calculated rate of 8.33% matches Option 1.

Step-by-Step Solution Summary

1. Identify that the money doubles, meaning the Amount (A) = 2 * Principal (P).
2. Calculate the Simple Interest (SI) earned: SI = A - P = 2P - P = P.
3. Identify the Time (T) = 12 years.
4. Use the simple interest formula: \(SI = \frac{P \times R \times T}{100}\).
5. Substitute the known values: \(P = \frac{P \times R \times 12}{100}\).
6. Solve for R: \(1 = \frac{R \times 12}{100} \implies 100 = 12R \implies R = \frac{100}{12}\).
7. Calculate the value of R: \(R = \frac{25}{3} \approx 8.33\).
8. Express the rate as a percentage: 8.33%.

Therefore, the rate percent per annum of simple interest required for a sum of money to double itself in 12 years is approximately 8.33%.

Revision Table: Simple Interest Concepts

Additional Information: Compound vs. Simple Interest

It's important to distinguish simple interest from compound interest.

• Simple Interest: Interest is calculated only on the original principal amount for the entire duration. The interest earned each period remains constant. This is what the question deals with.
• Compound Interest: Interest is calculated on the principal amount and also on the accumulated interest from previous periods. This means the interest earned grows over time because the base amount for calculating interest increases.

If the question involved compound interest, the calculation for the rate would be different. The formula for amount under compound interest is \(A = P \left(1 + \frac{R}{100}\right)^T\). If the money doubles (A=2P), then \(2P = P \left(1 + \frac{R}{100}\right)^{12}\), which simplifies to \(2 = \left(1 + \frac{R}{100}\right)^{12}\). Solving for R in this case would involve taking the 12th root, resulting in a different rate than simple interest.

Always pay close attention to whether the problem specifies simple interest or compound interest.