To find the sum invested, we can use the information provided to find the simple interest first, and then we can calculate the principal amount using the formula:

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

Step 1: Define the Known Variables

• Amount after 7 years, $A_1 = 14,522$
• Amount after 11 years, $A_2 = 18,906$
• Time difference, $T_2 - T_1 = 11 - 7 = 4$ years

Step 2: Calculate the Simple Interest Earned in the Last 4 Years

The increase in the amount over the last 4 years can be considered the simple interest earned for these 4 years. So, the simple interest for these 4 years is:

$\text{SI}_{4\text{ years}} = A_2 - A_1 = 18,906 - 14,522 = 4,384$

Step 3: Calculate the Annual Simple Interest

If $4,384$ is the interest for 4 years, then the interest for 1 year is:

$\text{SI}_{1\text{ year}} = \frac{4,384}{4} = 1,096$

Step 4: Calculate the Total Simple Interest for 7 Years

The simple interest for 7 years can be calculated using:

$\text{SI}_{7\text{ years}} = 1,096 \times 7 = 7,672$

Step 5: Calculate the Principal Amount

We know:

$\text{Amount} = \text{Principal} + \text{Simple Interest}$

Using the amount after 7 years:

$14,522 = P + 7,672$

$P = 14,522 - 7,672$

$P = 6,850$

Step 6: Verify by Finding the Total Amount After 11 Years

Now, using the principal amount to check if it aligns with the 11-year amount:

$\text{Total Interest for 11 years} = 1,096 \times 11 = 12,056$

$\text{Amount after 11 years} = P + \text{Total Interest for 11 years}$

$= 6,850 + 12,056 = 18,906 \, \text{(which matches the given amount after 11 years)}$

Therefore, the principal amount invested initially is 6,850.

Would you like more details or have any questions?

Related Questions:

1. How would the result change if the interest rate was different?
2. What would be the investment amount if the duration was only 5 years?
3. How can compound interest be calculated if the rate is known?
4. What if the investment increased by 10% each year, how would you calculate the total?
5. How does the simple interest formula change for different compounding intervals?

Tip: Always verify your calculated principal by checking it against different time frames provided in the problem to ensure accuracy.