If compound interest on some amount is 6500 for 15th year and 8651.2 for 18th year then find the rate of interest?

Options

A. `11%`
B. `9% `
C. `10%`
D. `8% `

Answer

To find the rate of interest based on the compound interest amounts given for the 15th and 18th years, we can follow these steps:

### Step 1: Understand the relationship between the years and the compound interest.
The compound interest for any year is calculated based on the principal amount and the interest rate. The interest for a particular year is calculated on the amount that includes the principal and the interest accumulated from previous years.

### Step 2: Set up the equations for the compound interest.
Let \( P \) be the principal amount and \( r \) be the rate of interest (in decimal).

- The compound interest for the 15th year can be expressed as:
\[ CI_{15} = P \times \left(1 + r\right)^{15} - P \times \left(1 + r\right)^{14} = P \times \left(1 + r\right)^{14} \times r \]
Given that \( CI_{15} = 6500 \), we can write:
\[ P \times \left(1 + r\right)^{14} \times r = 6500 \quad \text{(1)} \]

- The compound interest for the 18th year can be expressed as:
\[ CI_{18} = P \times \left(1 + r\right)^{18} - P \times \left(1 + r\right)^{17} = P \times \left(1 + r\right)^{17} \times r \]
Given that \( CI_{18} = 8651.2 \), we can write:
\[ P \times \left(1 + r\right)^{17} \times r = 8651.2 \quad \text{(2)} \]

### Step 3: Divide the two equations to eliminate \( P \).
Dividing equation (2) by equation (1):
\[ \frac{P \times \left(1 + r\right)^{17} \times r}{P \times \left(1 + r\right)^{14} \times r} = \frac{8651.2}{6500} \]
This simplifies to:
\[ \frac{\left(1 + r\right)^{17}}{\left(1 + r\right)^{14}} = \frac{8651.2}{6500} \]
\[ \left(1 + r\right)^{3} = \frac{8651.2}{6500} \]

### Step 4: Calculate the right side.
Calculating the fraction:
\[ \frac{8651.2}{6500} = 1.33 \]
So we have:
\[ \left(1 + r\right)^{3} = 1.33 \]

### Step 5: Take the cube root to find \( 1 + r \).
Taking the cube root:
\[ 1 + r = 1.33^{\frac{1}{3}} \]
Calculating the cube root:
\[ 1 + r \approx 1.1 \]

### Step 6: Solve for \( r \).
Thus:
\[ r \approx 1.1 - 1 = 0.1 \]
Converting \( r \) to a percentage:
\[ r \approx 0.1 \times 100 = 10\% \]

### Step 7: Conclusion.
The rate of interest is approximately **10%**.

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