The compound interest on a certain sum for 7th and 9th year are 'x' and 'y' respectively. Find the compound interest in 10th year?

Options

A. `sqrt ((x ^(2))/(y))`
B. `sqrt (( y ^(3))/(x))`
C. `sqrt (xy)`
D. `(y ^(2))/(x)`

Answer

To find the compound interest in the 10th year given the compound interest for the 7th year (x) and the 9th year (y), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Compound Interest**:
- The compound interest for the 7th year is represented as \( x \).
- The compound interest for the 9th year is represented as \( y \).

2. **Relationship Between Years**:
- The compound interest for the 8th year can be derived from the 7th and 9th years.
- The compound interest for the 8th year can be expressed as:
\[ \text{CI for 8th year} = \text{CI for 7th year} \times (1 + r/100) \]
- The compound interest for the 9th year can be expressed as:
\[ y = \text{CI for 8th year} \times (1 + r/100) = x \times (1 + r/100)^2 \]

3. **Finding the Growth Factor**:
- From the above relationship, we can derive:
\[ (1 + r/100)^2 = \frac{y}{x} \]
- Taking the square root of both sides gives:
\[ 1 + r/100 = \sqrt{\frac{y}{x}} \]

4. **Calculating Compound Interest for the 10th Year**:
- The compound interest for the 10th year can be expressed as:
\[ \text{CI for 10th year} = \text{CI for 9th year} \times (1 + r/100) = y \times (1 + r/100) \]
- Substituting \( 1 + r/100 \) from the previous step:
\[ \text{CI for 10th year} = y \times \sqrt{\frac{y}{x}} \]
- This simplifies to:
\[ \text{CI for 10th year} = \frac{y^{3/2}}{\sqrt{x}} \]

### Final Answer:
The compound interest in the 10th year is given by:
\[ \text{CI for 10th year} = \frac{y^{3/2}}{\sqrt{x}} \]