If a principal P becomes Q in 2 years when interest R% is compounded half-yearly. And if the same principal P becomes Q in 2 years when interest S% is compound annually, then which of the following is true?

Options

A. `RgtS`
B. `R=S`
C. `RltS`
D. `RleD`

Answer

To solve the problem step by step, we need to understand the relationship between the principal amount (P), the amount after 2 years (Q), the interest rates (R and S), and the compounding methods (half-yearly and annually).

### Step 1: Understanding the Compounding Formulas
1. **Half-Yearly Compounding**:
The formula for the amount A when interest is compounded half-yearly is given by:
\[ A = P \left(1 + \frac{R}{200}\right)^{2n} \]
where \( n \) is the number of years. For our case, \( n = 2 \), so:
\[ Q = P \left(1 + \frac{R}{200}\right)^{4} \]

2. **Annually Compounding**:
The formula for the amount A when interest is compounded annually is given by:
\[ A = P \left(1 + \frac{S}{100}\right)^{n} \]
For our case, \( n = 2 \), so:
\[ Q = P \left(1 + \frac{S}{100}\right)^{2} \]

### Step 2: Setting Up the Equations
From the above formulas, we have two equations:
1. For half-yearly compounding:
\[ Q = P \left(1 + \frac{R}{200}\right)^{4} \]
2. For annual compounding:
\[ Q = P \left(1 + \frac{S}{100}\right)^{2} \]

### Step 3: Equating the Two Expressions for Q
Since both expressions equal Q, we can set them equal to each other:
\[ P \left(1 + \frac{R}{200}\right)^{4} = P \left(1 + \frac{S}{100}\right)^{2} \]
We can cancel P from both sides (assuming P is not zero):
\[ \left(1 + \frac{R}{200}\right)^{4} = \left(1 + \frac{S}{100}\right)^{2} \]

### Step 4: Taking the Square Root
To simplify, we can take the square root of both sides:
\[ \left(1 + \frac{R}{200}\right)^{2} = \left(1 + \frac{S}{100}\right) \]

### Step 5: Expanding and Rearranging
Expanding the left side:
\[ 1 + \frac{R}{100} + \frac{R^2}{40000} = 1 + \frac{S}{100} \]
Subtracting 1 from both sides:
\[ \frac{R}{100} + \frac{R^2}{40000} = \frac{S}{100} \]

### Step 6: Analyzing the Relationship
From the above equation, we can see that:
\[ R + \frac{R^2}{400} = S \]
This implies that \( S \) is greater than \( R \) since \( \frac{R^2}{400} \) is a positive quantity for any positive R.

### Conclusion
Thus, we conclude that:
\[ R < S \]