Calculating the Principal using Simple Interest

This problem asks us to find the initial sum of money invested, also known as the principal, given the amount received after a certain period, the simple interest rate, and the time duration.

Let's break down the information given:

• Amount (A) = Rs. 834
• Simple Interest Rate (R) = 6% per annum
• Time (T) = \(6\frac{1}{2}\) years = 6.5 years

We need to find the Principal (P).

The formula for calculating Simple Interest (SI) is:

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

The Amount (A) is the sum of the Principal (P) and the Simple Interest (SI):

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

Substituting the formula for SI into the formula for A:

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

We can factor out P from the right side:

\( \text{A} = \text{P} \left( 1 + \frac{\text{R} \times \text{T}}{100} \right) \)

Now, we can plug in the given values:

\( 834 = \text{P} \left( 1 + \frac{6 \times 6.5}{100} \right) \)

Let's calculate the term inside the parenthesis:

\( \frac{6 \times 6.5}{100} = \frac{39}{100} = 0.39 \)

So the equation becomes:

\( 834 = \text{P} (1 + 0.39) \)

\( 834 = \text{P} (1.39) \)

To find P, we need to divide the Amount by 1.39:

\( \text{P} = \frac{834}{1.39} \)

Performing the division:

\( \text{P} = 600 \)

So, the initial sum invested was Rs. 600.

Step-by-Step Calculation of Principal

1. Identify the given values: Amount (A) = 834, Rate (R) = 6%, Time (T) = 6.5 years.
2. Recall the formula relating Amount, Principal, Rate, and Time under simple interest: \( \text{A} = \text{P} \left( 1 + \frac{\text{R} \times \text{T}}{100} \right) \).
3. Substitute the known values into the formula: \( 834 = \text{P} \left( 1 + \frac{6 \times 6.5}{100} \right) \).
4. Calculate the term \( \frac{\text{R} \times \text{T}}{100} \): \( \frac{6 \times 6.5}{100} = \frac{39}{100} = 0.39 \).
5. Simplify the equation: \( 834 = \text{P} (1 + 0.39) = \text{P} (1.39) \).
6. Solve for P by dividing the Amount by 1.39: \( \text{P} = \frac{834}{1.39} \).
7. Calculate the final value of P: \( \text{P} = 600 \).

The initial sum invested, the principal, was Rs. 600.

Revision Table: Simple Interest Concepts

Additional Information: Understanding Simple Interest

Simple interest is a fundamental concept in finance. It is the easiest way to calculate interest because it is based solely on the original principal amount. Unlike compound interest, where interest is added to the principal and earns interest itself, simple interest remains constant over the investment period, assuming the rate and principal don't change.

Key aspects of simple interest:

• The interest earned each period (e.g., each year) is the same.
• It is commonly used for short-term loans or simpler financial calculations.
• The total interest is directly proportional to the principal, rate, and time.

In this problem, knowing the final amount allowed us to work backward using the relationship between Amount, Principal, Rate, and Time in a simple interest scenario to find the original principal amount invested.