If some amount becomes 16000 in 4 years and 18522 in 7 years at certain rate of compound interset then find the rate of interset ?

Options

A. `7% `
B. `5% `
C. `8%`
D. `6%`

Answer

To find the rate of interest when an amount becomes 16,000 in 4 years and 18,522 in 7 years at a certain rate of compound interest, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest (in percent)
- \( A_1 \) = Amount after 4 years = 16,000
- \( A_2 \) = Amount after 7 years = 18,522

### Step 2: Write the Compound Interest Formulas
Using the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
We can write two equations based on the given amounts:

1. For 4 years:
\[ A_1 = P \left(1 + \frac{R}{100}\right)^4 = 16,000 \quad \text{(Equation 1)} \]

2. For 7 years:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^7 = 18,522 \quad \text{(Equation 2)} \]

### Step 3: Divide the Two Equations
To eliminate \( P \), we can divide Equation 2 by Equation 1:
\[ \frac{A_2}{A_1} = \frac{P \left(1 + \frac{R}{100}\right)^7}{P \left(1 + \frac{R}{100}\right)^4} \]
This simplifies to:
\[ \frac{18,522}{16,000} = \left(1 + \frac{R}{100}\right)^{7-4} \]
\[ \frac{18,522}{16,000} = \left(1 + \frac{R}{100}\right)^3 \]

### Step 4: Calculate the Left Side
Calculating the left side:
\[ \frac{18,522}{16,000} = 1.157625 \]
So we have:
\[ 1.157625 = \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Take the Cube Root
To find \( 1 + \frac{R}{100} \), we take the cube root of both sides:
\[ 1 + \frac{R}{100} = \sqrt[3]{1.157625} \]
Calculating the cube root:
\[ 1 + \frac{R}{100} \approx 1.05 \]

### Step 6: Solve for \( R \)
Now, we can solve for \( R \):
\[ \frac{R}{100} = 1.05 - 1 = 0.05 \]
\[ R = 0.05 \times 100 = 5 \]

### Conclusion
The rate of interest \( R \) is **5%**.

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