The compound interest on a sum for `4^(th)` year is Rs 6000 and compound interest for `5^(th)` year is Rs 6750(interest is compounded annually). What is the rate of interest?

Options

A. `25%`
B. `20%`
C. `12.5%`
D. `15%`

Answer

To solve the problem, we need to find the rate of interest based on the compound interest for the 4th and 5th years. Let's break it down step by step.

### Step 1: Understand the relationship between the years
The compound interest for the 4th year (CI4) is given as Rs 6000, and for the 5th year (CI5) it is Rs 6750. The compound interest for any year is calculated on the amount at the end of the previous year.

### Step 2: Set up the equations
Let A3 be the amount at the end of the 3rd year. Then:
- The amount at the end of the 4th year (A4) can be expressed as:
\[ A4 = A3 + CI4 = A3 + 6000 \]
- The amount at the end of the 5th year (A5) can be expressed as:
\[ A5 = A4 + CI5 = (A3 + 6000) + 6750 = A3 + 12750 \]

### Step 3: Relate the compound interest
The compound interest for the 5th year is also the interest on the amount at the end of the 4th year (A4):
\[ CI5 = A4 \times \frac{R}{100} \]
Where R is the rate of interest.

### Step 4: Substitute A4 in the equation
From the previous step, we know:
\[ CI5 = (A3 + 6000) \times \frac{R}{100} = 6750 \]

### Step 5: Set up the equation for the 4th year
Similarly, the compound interest for the 4th year can be expressed as:
\[ CI4 = A3 \times \frac{R}{100} = 6000 \]

### Step 6: Solve for A3
From the equation for CI4:
\[ A3 = \frac{6000 \times 100}{R} = \frac{600000}{R} \]

### Step 7: Substitute A3 in CI5 equation
Now substitute A3 in the CI5 equation:
\[ 6750 = \left(\frac{600000}{R} + 6000\right) \times \frac{R}{100} \]

### Step 8: Simplify the equation
Multiply both sides by 100 to eliminate the fraction:
\[ 675000 = \left(600000 + 6000R\right) \times \frac{R}{100} \]
\[ 675000 = \frac{600000R + 6000R^2}{100} \]
Multiply through by 100:
\[ 67500000 = 600000R + 6000R^2 \]

### Step 9: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ 6000R^2 + 600000R - 67500000 = 0 \]

### Step 10: Divide through by 6000
To simplify, divide everything by 6000:
\[ R^2 + 100R - 11250 = 0 \]

### Step 11: Solve the quadratic equation
Using the quadratic formula \(R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = 100\), and \(c = -11250\):
\[ R = \frac{-100 \pm \sqrt{100^2 - 4 \cdot 1 \cdot (-11250)}}{2 \cdot 1} \]
\[ R = \frac{-100 \pm \sqrt{10000 + 45000}}{2} \]
\[ R = \frac{-100 \pm \sqrt{55000}}{2} \]
Calculating \(\sqrt{55000}\):
\[ \sqrt{55000} \approx 234.5 \]
Thus:
\[ R = \frac{-100 + 234.5}{2} \approx \frac{134.5}{2} \approx 67.25 \]
This value is too high, so we check for possible errors or simplifications.

### Step 12: Check the options
Given the options, we can also find the rate by calculating:
\[ \text{Rate} = \frac{CI5 - CI4}{CI4} \times 100 \]
\[ \text{Rate} = \frac{6750 - 6000}{6000} \times 100 = \frac{750}{6000} \times 100 = 12.5\% \]

### Final Answer
The rate of interest is **12.5%**.