Understanding the Simple Interest Problem

This problem involves calculating the rate of simple interest per annum. We are given the total amount (principal plus simple interest) accumulated after two different time periods.

In simple interest, the interest earned each year is constant, based only on the original principal amount. Therefore, the increase in the amount over any period is solely due to the simple interest earned during that specific period.

Step-by-Step Calculation of Simple Interest

Let \(A_1\) be the amount after \(T_1\) years and \(A_2\) be the amount after \(T_2\) years. We are given:

• \(T_1 = 3\) years, \(A_1 = \text{Rs. } 3795\)
• \(T_2 = 7 \frac{1}{2} = 7.5\) years, \(A_2 = \text{Rs. } 4537.50\)

The difference in time is \(T_2 - T_1\):

\( \text{Time difference} = 7.5 - 3 = 4.5 \text{ years} \)

The difference in the amount is \(A_2 - A_1\):

\( \text{Amount difference} = 4537.50 - 3795 = 742.50 \)

This amount difference represents the simple interest earned over the 4.5 years between the two periods.

\( \text{Simple Interest for } 4.5 \text{ years} = \text{Rs. } 742.50 \)

Calculating Simple Interest per Year

Since simple interest is constant for each year, we can find the simple interest earned in one year by dividing the total interest for 4.5 years by 4.5.

\( \text{Simple Interest for 1 year} = \frac{\text{Simple Interest for } 4.5 \text{ years}}{\text{Time difference}} \)

\( \text{Simple Interest for 1 year} = \frac{742.50}{4.5} \)

To simplify the division, we can remove the decimals:

\( \frac{742.50}{4.5} = \frac{7425}{45} \)

Now, we perform the division:

\( 7425 \div 45 \)

We can simplify this fraction:

\( \frac{7425}{45} = \frac{1485}{9} = 165 \)

\( \text{Simple Interest for 1 year} = \text{Rs. } 165 \)

Finding the Principal Amount

The amount after 3 years is the principal plus the simple interest for 3 years. We know the simple interest for 1 year is Rs. 165.

\( \text{Simple Interest for 3 years} = \text{Simple Interest for 1 year} \times 3 \)

\( \text{Simple Interest for 3 years} = 165 \times 3 = \text{Rs. } 495 \)

The amount after 3 years is Rs. 3795. So, the Principal (P) is:

\( \text{Principal} = \text{Amount after 3 years} - \text{Simple Interest for 3 years} \)

\( P = 3795 - 495 = \text{Rs. } 3300 \)

Calculating the Rate of Interest

Now we have the Principal (P = Rs. 3300) and the Simple Interest for 1 year (SI = Rs. 165). The formula for simple interest is:

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

Where R is the rate of interest per annum and T is the time in years. Using T=1 year and SI=165:

\( 165 = \frac{3300 \times R \times 1}{100} \)

\( 165 = 33 \times R \)

To find R, divide 165 by 33:

\( R = \frac{165}{33} \)

\( R = 5 \)

The rate of interest is 5 percent per annum.

Revision Table: Simple Interest Formulas

Additional Information: Properties of Simple Interest

• Simple interest is calculated only on the initial principal amount.
• The simple interest earned each period (e.g., per year) remains constant throughout the loan or investment period.
• The amount grows linearly over time under simple interest. The difference in amounts between any two time periods is directly proportional to the time difference.
• Compound interest, unlike simple interest, calculates interest on the principal amount plus any accumulated interest.