Understanding the Simple Interest Problem

The problem asks us to find the original amount of money invested, also known as the principal, given the final amount received after a certain period at a specific simple interest rate. We are given the final amount (Principal + Simple Interest), the time period, and the annual simple interest rate.

Key Concepts and Formulas for Simple Interest

Simple interest is calculated only on the initial principal amount. The formulas we need are:

• Simple Interest (SI) = $ \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} $
• Amount (A) = Principal (P) + Simple Interest (SI)

Combining these, we can also write the Amount formula as:

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

or

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

Analyzing the Given Information

We are given the following values:

• Amount (A) = Rs. 732
• Time (T) = 2 years and 9 months
• Rate (R) = 8% per annum

We need to find the Principal (P).

Converting Time Period to Years

The time is given in years and months. We need to convert the entire time period into years to use it in the formula. There are 12 months in a year.

9 months = $ \frac{9}{12} $ years = $ \frac{3}{4} $ years = 0.75 years

Total Time (T) = 2 years + 0.75 years = 2.75 years

Calculating the Principal (P)

We use the formula for the Amount: $ \text{A} = \text{P} \left(1 + \frac{\text{R} \times \text{T}}{100}\right) $

Substitute the given values into the formula:

$ 732 = \text{P} \left(1 + \frac{8 \times 2.75}{100}\right) $

First, calculate the term inside the parenthesis:

$ 8 \times 2.75 = 22 $

$ \frac{8 \times 2.75}{100} = \frac{22}{100} = 0.22 $

Now, substitute this back into the equation:

$ 732 = \text{P} (1 + 0.22) $

$ 732 = \text{P} (1.22) $

To find P, divide the Amount by 1.22:

$ \text{P} = \frac{732}{1.22} $

To simplify the division, we can multiply both the numerator and denominator by 100 to remove the decimal:

$ \text{P} = \frac{732 \times 100}{1.22 \times 100} = \frac{73200}{122} $

Now, perform the division:

$ 73200 \div 122 $

We can see that $ 122 \times 6 = 732 $. Therefore, $ 122 \times 600 = 73200 $.

$ \text{P} = 600 $

So, the initial sum invested (Principal) was Rs. 600.

Verification

Let's check if a principal of Rs. 600 invested for 2.75 years at 8% simple interest per annum results in a final amount of Rs. 732.

$ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} = \frac{600 \times 8 \times 2.75}{100} $

$ \text{SI} = \frac{600 \times 22}{100} = 6 \times 22 = 132 $

Simple Interest = Rs. 132

Amount = Principal + Simple Interest = $ 600 + 132 = 732 $

This matches the given amount, so our calculated principal is correct.

Summary of Calculation Steps

The sum that was initially invested was Rs. 600.

Revision Table: Simple Interest Terms

Additional Information on Simple Interest Calculations

Understanding simple interest is fundamental before moving to compound interest. Here are some key points:

• Simple interest is easier to calculate compared to compound interest.
• The interest earned each year is the same if the principal, rate, and time unit (usually year) are constant.
• The formula $ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} $ is valid only when Rate (R) is the annual rate and Time (T) is in years. If time is given in months or days, it must be converted to years.
• If time is in months, convert to years by dividing by 12. If time is in days (using a standard year of 365 days), convert to years by dividing by 365.
• The relationship between Amount, Principal, and Simple Interest is always A = P + SI. This relationship is used to find any one value if the other two are known.