If some amount becomes 5500 in 3 years and 6050 in 4 years at compound interest then find the amount after six years?

Options

A. ₹7250.50
B. ₹7320.50
C. ₹7520.50
D. ₹7230.50

Answer

To solve the problem step by step, we will first find the rate of interest and then calculate the amount after 6 years.

### Step 1: Identify the given amounts
- Amount after 3 years (A3) = 5500
- Amount after 4 years (A4) = 6050

### Step 2: Calculate the interest for the 4th year
The interest earned in the 4th year can be calculated as:
\[ \text{Interest for 4th year} = A4 - A3 \]
Substituting the values:
\[ \text{Interest for 4th year} = 6050 - 5500 = 550 \]

### Step 3: Calculate the principal amount (P)
Using the amount after 3 years, we can express it in terms of the principal and the rate of interest (R):
\[ A3 = P(1 + \frac{R}{100})^3 \]
Using the amount after 4 years:
\[ A4 = P(1 + \frac{R}{100})^4 \]
From the two equations, we can find the rate of interest (R).

### Step 4: Find the rate of interest (R)
We know that:
\[ A4 = A3(1 + \frac{R}{100}) \]
Substituting the values:
\[ 6050 = 5500(1 + \frac{R}{100}) \]
Now, divide both sides by 5500:
\[ \frac{6050}{5500} = 1 + \frac{R}{100} \]
Calculating the left side:
\[ 1.1 = 1 + \frac{R}{100} \]
Now, subtract 1 from both sides:
\[ 0.1 = \frac{R}{100} \]
Multiplying both sides by 100 gives:
\[ R = 10\% \]

### Step 5: Calculate the amount after 6 years (A6)
Using the principal amount (P) and the rate of interest (R), we can find the amount after 6 years:
\[ A6 = P(1 + \frac{R}{100})^6 \]
We need to find P first. Using the amount after 3 years:
\[ 5500 = P(1 + \frac{10}{100})^3 \]
This simplifies to:
\[ 5500 = P(1.1)^3 \]
Calculating \( (1.1)^3 \):
\[ (1.1)^3 = 1.331 \]
Now substituting back:
\[ 5500 = P \times 1.331 \]
To find P:
\[ P = \frac{5500}{1.331} \approx 4134.16 \]

### Step 6: Substitute P into the formula for A6
Now we can calculate A6:
\[ A6 = 4134.16(1.1)^6 \]
Calculating \( (1.1)^6 \):
\[ (1.1)^6 \approx 1.771561 \]
Now substituting back:
\[ A6 \approx 4134.16 \times 1.771561 \approx 7315.72 \]

### Final Answer
The amount after 6 years is approximately **7315.72**.

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