Find the ratio of compound interest to simple interest on a sum at `8%` per annum for 3 years

Options

A. `1051:1529`
B. `129:427`
C. `2:3`
D. `2029:1875`

Answer

To find the ratio of compound interest (CI) to simple interest (SI) on a sum at 8% per annum for 3 years, we can follow these steps:

### Step 1: Calculate the Compound Interest (CI)
1. **Formula for Amount (A) in Compound Interest**:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \(P\) = Principal amount (assume \(P = 100\))
- \(r\) = Rate of interest (8%)
- \(t\) = Time (3 years)

2. **Substituting the values**:
\[ A = 100 \left(1 + \frac{8}{100}\right)^3 = 100 \left(1 + 0.08\right)^3 = 100 \left(1.08\right)^3 \]

3. **Calculate \( (1.08)^3 \)**:
\[ (1.08)^3 = 1.08 \times 1.08 \times 1.08 = 1.259712 \]
Thus,
\[ A = 100 \times 1.259712 = 125.9712 \]

4. **Calculate Compound Interest (CI)**:
\[ CI = A - P = 125.9712 - 100 = 25.9712 \]

### Step 2: Calculate the Simple Interest (SI)
1. **Formula for Simple Interest (SI)**:
\[ SI = \frac{P \times r \times t}{100} \]

2. **Substituting the values**:
\[ SI = \frac{100 \times 8 \times 3}{100} = \frac{2400}{100} = 24 \]

### Step 3: Calculate the Ratio of CI to SI
1. **Ratio of Compound Interest to Simple Interest**:
\[ \text{Ratio} = \frac{CI}{SI} = \frac{25.9712}{24} \]

2. **Simplifying the ratio**:
- To eliminate the decimal, multiply both the numerator and denominator by 100:
\[ \text{Ratio} = \frac{2597.12}{2400} \]

3. **Calculating the ratio**:
- Dividing both by 8:
\[ \frac{2597.12 \div 8}{2400 \div 8} = \frac{324.64}{300} \approx \frac{81.16}{75} \approx \frac{2029}{1875} \]

### Final Answer
The ratio of compound interest to simple interest is:
\[ \text{Ratio} = \frac{2029}{1875} \]