If simple interest on some amount in 4 year is 500 and difference between simple interest and compound interest for 2 years is 25 then find the rate of interest?

Options

A. `25%`
B. `4.5%`
C. `5.5%`
D. `20%`

Answer

To solve the problem step by step, we will use the formulas for Simple Interest (SI) and Compound Interest (CI).

### Step 1: Understand the given information
We know:
- Simple Interest (SI) for 4 years = 500
- Difference between Compound Interest (CI) and Simple Interest (SI) for 2 years = 25

### Step 2: Use the formula for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \cdot R \cdot T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal amount
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

Given that \(SI = 500\) and \(T = 4\):
\[ 500 = \frac{P \cdot R \cdot 4}{100} \]
This simplifies to:
\[ 500 = \frac{4PR}{100} \]
Multiplying both sides by 100:
\[ 50000 = 4PR \]
Dividing by 4:
\[ PR = 12500 \quad \text{(Equation 1)} \]

### Step 3: Understand the difference between CI and SI
The difference between Compound Interest and Simple Interest for 2 years can be expressed as:
\[ CI - SI = \frac{P \cdot R^2}{100^2} \]
Given that this difference is 25:
\[ \frac{P \cdot R^2}{100^2} = 25 \]
Multiplying both sides by \(100^2\):
\[ P \cdot R^2 = 250000 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we have \(P = \frac{12500}{R}\). Substitute this into Equation 2:
\[ \left(\frac{12500}{R}\right) \cdot R^2 = 250000 \]
This simplifies to:
\[ 12500R = 250000 \]
Dividing both sides by 12500:
\[ R = 20 \]

### Step 5: Conclusion
The rate of interest \(R\) is 20%.

### Final Answer
The required rate of interest is **20%**.

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