Find the ratio of simple interest to compound interest for 2 years at 4% per annum, compounded yearly in case of compound interest.

Options

A. `50:53`
B. `50:51`
C. `49:50`
D. `48:53`

Answer

To find the ratio of simple interest (SI) to compound interest (CI) for 2 years at an interest rate of 4% per annum, compounded yearly, we can follow these steps:

### Step 1: Understand the formulas
- **Simple Interest (SI)** is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \(P\) is the principal amount, \(R\) is the rate of interest per annum, and \(T\) is the time in years.

- **Compound Interest (CI)** is calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \(A\) is the total amount after time \(T\). The compound interest can then be found by:
\[ CI = A - P \]

### Step 2: Assume a principal amount
Let's assume a principal amount \(P = 100\) for easier calculation.

### Step 3: Calculate Simple Interest for 2 years
Using the SI formula:
- \(R = 4\%\)
- \(T = 2\)

\[ SI = \frac{100 \times 4 \times 2}{100} = 8 \]

### Step 4: Calculate Compound Interest for 2 years
Using the CI formula:
\[ A = 100 \left(1 + \frac{4}{100}\right)^2 = 100 \left(1 + 0.04\right)^2 = 100 \left(1.04\right)^2 \]

Calculating \(1.04^2\):
\[ 1.04^2 = 1.0816 \]
Thus,
\[ A = 100 \times 1.0816 = 108.16 \]

Now, calculate the Compound Interest:
\[ CI = A - P = 108.16 - 100 = 8.16 \]

### Step 5: Find the ratio of Simple Interest to Compound Interest
Now we have:
- \(SI = 8\)
- \(CI = 8.16\)

The ratio of SI to CI is:
\[ \text{Ratio} = \frac{SI}{CI} = \frac{8}{8.16} \]

To simplify this ratio:
\[ \text{Ratio} = \frac{8}{8.16} \approx 0.9804 \]

### Step 6: Express the ratio in simpler terms
To express this in a more manageable form, we can multiply both the numerator and denominator by 100:
\[ \text{Ratio} \approx \frac{800}{816} = \frac{100}{102} = \frac{50}{51} \]

### Final Answer
The ratio of Simple Interest to Compound Interest for 2 years at 4% per annum is:
\[ \frac{50}{51} \]

---