If the difference between CI and SI on 1000 for 2 years is 22.50 then find the rate of interest?

Options

A. `15%`
B. `16%`
C. `14% `
D. `17% `

Answer

To solve the problem step-by-step, we will find the rate of interest (R) given that the difference between Compound Interest (CI) and Simple Interest (SI) on a principal amount of 1000 for 2 years is 22.50.

### Step 1: Understand the relationship between CI and SI
The difference between CI and SI for 2 years can be expressed as:
\[ \text{Difference} = CI - SI \]

### Step 2: Write the formulas for CI and SI
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P = 1000 \) (the principal)
- \( R \) = rate of interest
- \( T = 2 \) (time in years)

So,
\[ SI = \frac{1000 \times R \times 2}{100} = 20R \]

- The formula for Compound Interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values:
\[ CI = 1000 \left(1 + \frac{R}{100}\right)^2 - 1000 \]

### Step 3: Expand the CI formula
Using the binomial expansion for \( \left(1 + \frac{R}{100}\right)^2 \):
\[ CI = 1000 \left(1 + \frac{2R}{100} + \frac{R^2}{10000}\right) - 1000 \]
\[ CI = 1000 + 20R + \frac{1000R^2}{10000} - 1000 \]
\[ CI = 20R + \frac{R^2}{10} \]

### Step 4: Set up the equation based on the difference
According to the problem, the difference between CI and SI is given as 22.50:
\[ CI - SI = 22.50 \]
Substituting the values we derived:
\[ \left(20R + \frac{R^2}{10}\right) - 20R = 22.50 \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{R^2}{10} = 22.50 \]

### Step 6: Multiply both sides by 10 to eliminate the fraction
\[ R^2 = 22.50 \times 10 \]
\[ R^2 = 225 \]

### Step 7: Take the square root to find R
\[ R = \sqrt{225} \]
\[ R = 15 \]

### Conclusion
The rate of interest \( R \) is **15%**.

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