Harish borrowed certain sum from Harsh for 2 years at SI. Harish lent this sum to Dinesh at the same rate for 2 years at Compound Interest. At the end of second year, Harish received Rs 550 as compound interest but paid Rs 500 as simple interest. Find the rate of interest

Options

A. 25%
B. 20%
C. 15%
D. 22.5%

Answer

To solve the problem step by step, let's break down the information given and use the formulas for Simple Interest (SI) and Compound Interest (CI).

### Step 1: Understand the given information
- Harish borrowed a certain sum (let's call it P) from Harsh for 2 years at Simple Interest.
- Harish lent this sum to Dinesh at the same rate for 2 years at Compound Interest.
- Harish received Rs 550 as Compound Interest (CI) from Dinesh.
- Harish paid Rs 500 as Simple Interest (SI) to Harsh.

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

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

### Step 3: Set up the equations
1. From the Simple Interest:
\[ 500 = \frac{P \times R \times 2}{100} \]
Simplifying this gives:
\[ P \times R = 25000 \quad \text{(Equation 1)} \]

2. From the Compound Interest:
\[ 550 = P \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \]
Expanding this gives:
\[ 550 = P \left( \frac{R^2}{10000} + \frac{2R}{100} \right) \]
Rearranging gives:
\[ 550 = P \left( \frac{R^2 + 200R}{10000} \right) \]
Multiplying through by 10000:
\[ 5500000 = P(R^2 + 200R) \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we can express P in terms of R:
\[ P = \frac{25000}{R} \]
Substituting this into Equation 2:
\[ 5500000 = \frac{25000}{R}(R^2 + 200R) \]
Multiplying both sides by R:
\[ 5500000R = 25000(R^2 + 200R) \]
Expanding gives:
\[ 5500000R = 25000R^2 + 5000000R \]
Rearranging terms:
\[ 0 = 25000R^2 - 500000R + 5500000R \]
This simplifies to:
\[ 25000R^2 - 500000R + 5500000 = 0 \]

### Step 5: Solve the quadratic equation
Dividing the entire equation by 25000:
\[ R^2 - 20R + 220 = 0 \]
Using the quadratic formula \( R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = -20, c = 220 \)
\[ R = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 220}}{2 \cdot 1} \]
Calculating the discriminant:
\[ R = \frac{20 \pm \sqrt{400 - 880}}{2} \]
\[ R = \frac{20 \pm \sqrt{-480}}{2} \]
Since the discriminant is negative, we need to check our calculations again.

### Step 6: Check the ratio of CI to SI
From the problem, we can also use the ratio of CI to SI:
\[ \frac{CI}{SI} = \frac{550}{500} = \frac{11}{10} \]
Using the relationship:
\[ \frac{CI}{SI} = \frac{200 + R}{200} \]
Setting up the equation:
\[ \frac{11}{10} = \frac{200 + R}{200} \]
Cross-multiplying gives:
\[ 11 \cdot 200 = 10(200 + R) \]
\[ 2200 = 2000 + 10R \]
\[ 200 = 10R \]
\[ R = 20 \]

### Conclusion
The rate of interest is **20%**.

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