Understanding the Simple Interest Problem

The question asks us to find the rate of simple interest per annum. We are given that a certain sum of money (the Principal) becomes \(\frac{3}{2}\) times itself (this is the Amount) in 2 years when simple interest is applied.

Let's define the key terms in this simple interest problem:

• Principal (P): The initial sum of money.
• Amount (A): The total sum of money after adding the simple interest to the principal.
• Simple Interest (SI): The interest earned only on the principal amount over a period of time.
• Rate of Simple Interest (R): The percentage at which interest is calculated per annum.
• Time (T): The duration for which the money is lent or borrowed, in years.

According to the question:

• The Amount (A) is \(\frac{3}{2}\) of the Principal (P). So, \(A = \frac{3}{2}P\).
• The Time (T) is 2 years.
• We need to find the Rate of Simple Interest (R).

Calculating the Simple Interest Earned

The Amount is the sum of the Principal and the Simple Interest:

\(A = P + SI\)

We know \(A = \frac{3}{2}P\), so we can substitute this into the formula:

\(\frac{3}{2}P = P + SI\)

To find the Simple Interest (SI), we can rearrange the equation:

\(SI = \frac{3}{2}P - P\)

\(SI = (\frac{3}{2} - 1)P\)

\(SI = (\frac{3 - 2}{2})P\)

\(SI = \frac{1}{2}P\)

So, the Simple Interest earned is half of the Principal amount.

Finding the Rate of Simple Interest per Annum

The formula for Simple Interest is:

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

We have \(SI = \frac{1}{2}P\) and \(T = 2\) years. Let's substitute these values into the formula:

\(\frac{1}{2}P = \frac{P \times R \times 2}{100}\)

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

\(\frac{1}{2} = \frac{R \times 2}{100}\)

Now, we need to solve for R. Multiply both sides by 100:

\(\frac{1}{2} \times 100 = R \times 2\)

\(50 = 2R\)

Divide both sides by 2:

\(R = \frac{50}{2}\)

\(R = 25\)

The rate of simple interest is 25% per annum.

Let's summarise the steps:

1. Identify the given information: Amount is \(\frac{3}{2}\) times the Principal, Time is 2 years.
2. Understand the relationship between Amount, Principal, and Simple Interest: \(A = P + SI\).
3. Calculate the Simple Interest (SI) in terms of Principal (P) using the relationship \(A = \frac{3}{2}P\).
4. Use the Simple Interest formula: \(SI = \frac{P \times R \times T}{100}\).
5. Substitute the known values of SI, P, and T into the formula.
6. Solve the equation to find the rate of simple interest (R).

This calculation confirms that the rate of simple interest is 25%.

Revision Table: Key Simple Interest Formulas

Additional Information on Simple Interest Calculations

Simple interest is one of the most basic concepts in finance. It is calculated only on the initial principal amount. This means the interest earned in each period (e.g., each year) is constant, provided the principal and rate remain unchanged. This is different from compound interest, where interest is calculated on the initial principal as well as the accumulated interest from previous periods.

When solving simple interest problems, always ensure that the time period (T) and the rate (R) are in corresponding units. If the rate is per annum (yearly), the time should be in years. If the time is given in months, convert it to years by dividing by 12. If it's in days, divide by 365 (usually, unless specified otherwise).

In this specific problem, the time was already given in years, making the calculation straightforward using the simple interest formula.