A sum of 16,000, invested at simple interest, amounts to 22,400 in 4 years at a certain rate of interest. If the same sum of money is invested for 2 years at the same rate of interest, compounded p.a., find the compound interest earned.

Answer

To solve the problem step by step, we will first determine the rate of interest using the simple interest formula, and then we will calculate the compound interest for the given time period.

### Step 1: Identify the given values
- Principal (P) = 16,000
- Amount after 4 years (A) = 22,400
- Time (T) = 4 years
- Rate of interest (R) = ?

### Step 2: Calculate the simple interest (SI)
The formula for simple interest is:
\[ \text{SI} = A - P \]
Substituting the values:
\[ \text{SI} = 22,400 - 16,000 = 6,400 \]

### Step 3: Use the simple interest formula to find the rate of interest
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 6,400 = \frac{16,000 \times R \times 4}{100} \]

### Step 4: Simplify the equation to find R
First, multiply both sides by 100:
\[ 640,000 = 16,000 \times R \times 4 \]
Now, divide both sides by \( 16,000 \times 4 \):
\[ R = \frac{640,000}{64,000} = 10\% \]

### Step 5: Calculate the compound interest for 2 years
Now we will use the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- P = 16,000
- R = 10%
- T = 2 years

Substituting the values:
\[ A = 16,000 \left(1 + \frac{10}{100}\right)^2 \]
\[ A = 16,000 \left(1 + 0.1\right)^2 \]
\[ A = 16,000 \left(1.1\right)^2 \]
\[ A = 16,000 \times 1.21 = 19,360 \]

### Step 6: Calculate the compound interest (CI)
To find the compound interest, subtract the principal from the amount:
\[ \text{CI} = A - P \]
\[ \text{CI} = 19,360 - 16,000 = 3,360 \]

### Final Answer
The compound interest earned is **3,360**.

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