Investment 14522in7 year and 18906in11 year find sum invest

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.

Learn how to calculate the sum invested over different time periods using the simple interest formula. Solve a problem involving investments over 7 and 11 years, determining the initial principal amount based on given amounts after each period. Suitable for high school students studying financial mathematics.

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