To solve the problem step by step, we will follow the outlined approach to find the compound interest on a sum of Rs. 6,550 in 4 years at the same rate of interest derived from the first part of the question.
### Step 1: Find the Rate of Interest
We know:
- Principal (P) = Rs. 5,000
- Amount (A) = Rs. 7,200
- Time (T) = 8 years
Using the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values:
\[ 7,200 = 5,000 \left(1 + \frac{R}{100}\right)^8 \]
### Step 2: Simplify the Equation
To isolate the term with R, we divide both sides by 5,000:
\[ \frac{7,200}{5,000} = \left(1 + \frac{R}{100}\right)^8 \]
Calculating the left side:
\[ 1.44 = \left(1 + \frac{R}{100}\right)^8 \]
### Step 3: Take the Eighth Root
To solve for \(1 + \frac{R}{100}\), we take the eighth root of both sides:
\[ 1 + \frac{R}{100} = (1.44)^{\frac{1}{8}} \]
Calculating \( (1.44)^{\frac{1}{8}} \):
Using a calculator, we find:
\[ 1 + \frac{R}{100} \approx 1.043 \]
### Step 4: Solve for R
Now, we can isolate R:
\[ \frac{R}{100} \approx 1.043 - 1 \]
\[ \frac{R}{100} \approx 0.043 \]
\[ R \approx 4.3\% \]
### Step 5: Calculate the Compound Interest for Rs. 6,550
Now we will use the same rate of interest to find the compound interest on Rs. 6,550 for 4 years.
Using the formula again:
- Principal (P) = Rs. 6,550
- Time (T) = 4 years
- Rate (R) = 4.3%
The amount (A) can be calculated as:
\[ A = 6,550 \left(1 + \frac{4.3}{100}\right)^4 \]
### Step 6: Calculate the Amount
Calculating the term inside the parentheses:
\[ 1 + \frac{4.3}{100} = 1.043 \]
Now substituting back into the amount formula:
\[ A = 6,550 \times (1.043)^4 \]
Calculating \( (1.043)^4 \):
Using a calculator, we find:
\[ (1.043)^4 \approx 1.184 \]
Now substituting this value:
\[ A \approx 6,550 \times 1.184 \approx 7,760.20 \]
### Step 7: Calculate the Compound Interest
Now, we find the compound interest (CI):
\[ CI = A - P \]
\[ CI = 7,760.20 - 6,550 \]
\[ CI \approx 1,210.20 \]
### Final Answer
The compound interest on a sum of Rs. 6,550 in 4 years at the same rate of interest is approximately **Rs. 1,210.20**.
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