Understanding the Simple Interest Problem

The question asks us to find the annual rate of simple interest applied to an investment. We are given the initial investment amount, the time period, and the final amount received after the investment period.

Identifying the Given Values

We are provided with the following information:

• Principal Amount (P): The initial amount invested by Disha is Rs. 20,000.
• Time Period (T): The duration for which the amount was invested is 9 years.
• Total Amount (A): The final amount received after 9 years is Rs. 33,500.

We need to calculate the Rate of Interest (R) per annum.

Calculating the Simple Interest Earned

The total amount received is the sum of the principal amount and the simple interest earned over the time period. Therefore, we can find the simple interest (SI) by subtracting the principal from the total amount.

The formula for Simple Interest is:

\(\text{Simple Interest (SI)} = \text{Total Amount (A)} - \text{Principal Amount (P)}\)

Substituting the given values:

\(\text{SI} = \text{Rs. } 33,500 - \text{Rs. } 20,000\)

\(\text{SI} = \text{Rs. } 13,500\)

So, the simple interest earned over 9 years is Rs. 13,500.

Finding the Simple Interest Rate

The formula relating Simple Interest, Principal, Rate, and Time is:

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

We need to find the Rate (R). We can rearrange the formula to solve for R:

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

Now, we substitute the values we know:

• SI = Rs. 13,500
• P = Rs. 20,000
• T = 9 years

Substituting these into the formula for R:

\(\text{R} = \frac{13,500 \times 100}{20,000 \times 9}\)

\(\text{R} = \frac{1,350,000}{180,000}\)

We can simplify this expression by cancelling out common zeros:

\(\text{R} = \frac{1350}{180}\)

\(\text{R} = \frac{135}{18}\)

Both 135 and 18 are divisible by 9:

\(\text{R} = \frac{135 \div 9}{18 \div 9} = \frac{15}{2}\)

Converting the fraction to a decimal:

\(\text{R} = 7.5\)

Since R is the rate as a percentage, the rate of interest is 7.5% per annum.

Summary of Calculation Steps

Conclusion on the Rate of Interest

Based on the calculations, the annual rate of simple interest is 7.5%.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Calculations

Simple interest is a straightforward way to calculate interest on a principal amount. It is most commonly used for short-term loans or investments. Unlike compound interest, simple interest does not involve adding the earned interest back to the principal for subsequent calculations. The interest is always calculated on the original principal amount.

Understanding the relationship between Principal, Rate, Time, Simple Interest, and Amount is crucial for solving simple interest problems. By rearranging the basic formula, you can find any missing variable if the others are known.

For example, if you knew the Simple Interest, Rate, and Time, you could find the Principal using the formula:

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

Similarly, you could find the Time:

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

These rearrangements are derived directly from the main simple interest formula \(SI = \frac{P \times R \times T}{100}\).