Calculating the Sum Invested with Simple Interest

This problem involves simple interest, where the interest earned each year is constant and is calculated only on the original principal amount. We are given the amounts accumulated after two different time periods and need to find the initial sum invested, also known as the principal.

Understanding Simple Interest Accumulation

Under simple interest, the amount (A) after T years is given by the formula:

$$A = P + SI$$

Where P is the principal and SI is the simple interest earned. The simple interest for T years is:

$$SI = \frac{P \times R \times T}{100}$$

Here, R is the rate of interest per annum. Therefore, the amount can also be written as:

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

The key property of simple interest is that the interest earned for each year is the same.

Step-by-Step Solution to Find the Sum Invested

Let P be the sum invested (principal) and R be the rate of simple interest per annum.

Amounts Given:

• Amount after 7 years: Rs. 14,522
• Amount after 11 years: Rs. 18,906

Formulating Equations:

Based on the information, we can write two equations:

Amount after 7 years = Principal + Simple Interest for 7 years

$$14522 = P + SI_7 \quad (Equation\; 1)$$

Amount after 11 years = Principal + Simple Interest for 11 years

$$18906 = P + SI_{11} \quad (Equation\; 2)$$

Finding Interest Earned in the Difference Period:

The difference between the amounts after 11 years and 7 years is purely the simple interest earned during these (11 - 7) = 4 years.

Simple Interest for (11 - 7) years = Amount after 11 years - Amount after 7 years

Simple Interest for 4 years = $$18906 - 14522$$

Simple Interest for 4 years = $$4384$$

Finding Annual Simple Interest:

Since the simple interest is constant per year, we can find the interest earned in 1 year by dividing the interest for 4 years by 4.

Simple Interest for 1 year = $$\frac{4384}{4}$$

Simple Interest for 1 year = $$1096$$

Finding Simple Interest for 7 Years:

Now we can find the total simple interest earned in 7 years by multiplying the annual interest by 7.

Simple Interest for 7 years = Simple Interest for 1 year $$\times$$ 7

Simple Interest for 7 years = $$1096 \times 7$$

Simple Interest for 7 years = $$7672$$

Calculating the Principal (Sum Invested):

We can now use Equation 1 (Amount after 7 years = Principal + Simple Interest for 7 years) to find the principal P.

$$14522 = P + 7672$$

To find P, subtract the simple interest for 7 years from the amount after 7 years.

$$P = 14522 - 7672$$

$$P = 6850$$

Thus, the sum invested is Rs. 6850.

Verification (Optional):

We can verify this by calculating the simple interest for 11 years and checking if it matches with the amount after 11 years.

Simple Interest for 11 years = Simple Interest for 1 year $$\times$$ 11

Simple Interest for 11 years = $$1096 \times 11$$

Simple Interest for 11 years = $$12056$$

Amount after 11 years = Principal + Simple Interest for 11 years

Amount after 11 years = $$6850 + 12056$$

Amount after 11 years = $$18906$$

This matches the given amount after 11 years, confirming our calculated principal is correct.

Conclusion

By finding the interest earned over the difference in time periods, we could calculate the annual simple interest and subsequently the total interest for 7 years. Subtracting this interest from the amount after 7 years gave us the original sum invested, which is Rs. 6850.

Revision Table: Simple Interest Concepts

Additional Information: Simple vs. Compound Interest

It is important to distinguish between simple interest and compound interest.

• Simple Interest: Interest is calculated only on the original principal. The interest earned does not get added back to the principal for calculating future interest. This is why the simple interest amount for each year remains constant.
• Compound Interest: Interest is calculated on the principal amount plus any accumulated interest from previous periods. The interest is 'compounded', meaning it earns interest on interest. The amount grows faster under compound interest compared to simple interest for the same principal, rate, and time (for T > 1 year).

In this problem, the constant annual interest difference clearly indicates that simple interest is being applied.