Calculating Original Sum with Simple Interest Rates

The question asks us to find the initial sum of money (the principal) invested, given the total simple interest earned over a period of 18 years, during which the interest rate changed multiple times.

Simple interest is calculated using the formula:

Simple Interest (SI) = \frac{Principal (P) \times Rate (R) \times Time (T)}{100}

In this problem, the total time is divided into three periods with different interest rates:

• Period 1: 8 years at 7.25% per annum
• Period 2: 6 years at 8.5% per annum
• Period 3: 4 years at 6.5% per annum

The total duration is $8 + 6 + 4 = 18$ years, which matches the total time mentioned.

Let the original sum invested be P (in Rs.). We can calculate the simple interest earned in each period and sum them up to get the total interest.

Simple interest for the first 8 years ($SI_1$) at 7.25%:

SI_1 = \frac{P \times 7.25 \times 8}{100}

Simple interest for the next 6 years ($SI_2$) at 8.5%:

SI_2 = \frac{P \times 8.5 \times 6}{100}

Simple interest for the final 4 years ($SI_3$) at 6.5%:

SI_3 = \frac{P \times 6.5 \times 4}{100}

The total simple interest earned over 18 years is given as Rs. 35,100. So, the sum of the interests from the three periods must equal this amount:

Total Interest = SI_1 + SI_2 + SI_3 = 35100

Substituting the expressions for $SI_1$, $SI_2$, and $SI_3$:

\frac{P \times 7.25 \times 8}{100} + \frac{P \times 8.5 \times 6}{100} + \frac{P \times 6.5 \times 4}{100} = 35100

Calculate the products of rate and time for each period:

• $7.25 \times 8 = 58$
• $8.5 \times 6 = 51$
• $6.5 \times 4 = 26$

Now substitute these values back into the equation:

\frac{P \times 58}{100} + \frac{P \times 51}{100} + \frac{P \times 26}{100} = 35100

\frac{58P}{100} + \frac{51P}{100} + \frac{26P}{100} = 35100

Combine the terms on the left side:

\frac{(58 + 51 + 26)P}{100} = 35100

\frac{135P}{100} = 35100

Now, solve for P:

135P = 35100 \times 100

135P = 3510000

P = \frac{3510000}{135}

Let's perform the division:

P = 26000

So, the original sum invested was Rs. 26,000.

Let's verify the interest calculation with P = 26000:

• $SI_1 = \frac{26000 \times 7.25 \times 8}{100} = \frac{26000 \times 58}{100} = 260 \times 58 = 15080$
• $SI_2 = \frac{26000 \times 8.5 \times 6}{100} = \frac{26000 \times 51}{100} = 260 \times 51 = 13260$
• $SI_3 = \frac{26000 \times 6.5 \times 4}{100} = \frac{26000 \times 26}{100} = 260 \times 26 = 6760$

Total Interest = $SI_1 + SI_2 + SI_3 = 15080 + 13260 + 6760 = 35100$.

The calculated total interest matches the given total interest (Rs. 35,100). Therefore, the calculated principal amount is correct.

The original sum invested was Rs. 26,000.

Revision Table: Key Simple Interest Concepts

Additional Information on Varying Interest Rates

When the simple interest rate changes over the total period of investment or loan, the total simple interest is the sum of the simple interest calculated for each period with its respective rate and duration.

If the principal is P, and the rates are $R_1, R_2, R_3, \dots, R_n$ for periods $T_1, T_2, T_3, \dots, T_n$ respectively (where the total time is $T = T_1 + T_2 + \dots + T_n$), the total simple interest is:

Total SI = $\frac{P \times R_1 \times T_1}{100} + \frac{P \times R_2 \times T_2}{100} + \dots + \frac{P \times R_n \times T_n}{100}$

This can be simplified as:

Total SI = $\frac{P}{100} \times (R_1 T_1 + R_2 T_2 + \dots + R_n T_n)$

In this problem, we used this principle to set up the equation and solve for the unknown principal P.