At what rate of compound interest per annum will a sum of Rs.1,400 earn Rs.173.04 in 2 years?

Options

A. `4%`
B. `5%`
C. `6%`
D. `8%`

Answer

To find the rate of compound interest per annum at which a sum of Rs. 1,400 will earn Rs. 173.04 in 2 years, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 1,400
- Compound Interest (CI) = Rs. 173.04
- Time (T) = 2 years

### Step 2: Calculate the total amount (A)
The total amount (A) after 2 years can be calculated using the formula:
\[ A = P + CI \]
Substituting the values:
\[ A = 1400 + 173.04 = 1573.04 \]

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- A = total amount after time T
- P = principal amount
- R = rate of interest per annum
- T = time in years

### Step 4: Substitute the known values into the formula
Substituting the values we have:
\[ 1573.04 = 1400 \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Simplify the equation
To isolate the term with R, divide both sides by 1400:
\[ \frac{1573.04}{1400} = \left(1 + \frac{R}{100}\right)^2 \]
Calculating the left side:
\[ \frac{1573.04}{1400} = 1.1236 \]
So we have:
\[ 1.1236 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 6: Take the square root of both sides
Taking the square root gives:
\[ \sqrt{1.1236} = 1 + \frac{R}{100} \]
Calculating the square root:
\[ 1.06 = 1 + \frac{R}{100} \]

### Step 7: Solve for R
Now, subtract 1 from both sides:
\[ 1.06 - 1 = \frac{R}{100} \]
This simplifies to:
\[ 0.06 = \frac{R}{100} \]
Multiplying both sides by 100 gives:
\[ R = 6 \]

### Conclusion
The rate of compound interest per annum is **6%**.

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