What is the rate of interest pcpa ? Statements: I. The difference between the compound interest and simple interest earned in two years on the amount invested is Rs.100. II. The amount becomes Rs.19,500 in three years on simple interest. III. The simple interest acquired in two years on the same amount at the same rate of interest is `Rs. 3,000.

Options

A. only I and II
B. only I and III
C. only II and III
D. only I and either II or III

Explanation

To find the rate of interest per annum (PCPA), we will analyze the statements provided step by step.

### Step 1: Analyze Statement I
Statement I states that the difference between the compound interest (CI) and simple interest (SI) earned in two years on the amount invested is Rs. 100.

The formula for the difference between CI and SI for two years is:
\[ \text{Difference} = \frac{P \cdot R^2}{100^2} \]
where \( P \) is the principal amount and \( R \) is the rate of interest.

From Statement I, we have:
\[ \frac{P \cdot R^2}{100^2} = 100 \]
This can be rewritten as:
\[ P \cdot R^2 = 100 \cdot 100^2 \]
\[ P \cdot R^2 = 10000 \quad \text{(Equation 1)} \]

### Step 2: Analyze Statement II
Statement II states that the amount becomes Rs. 19,500 in three years on simple interest.

The formula for the amount \( A \) on simple interest is:
\[ A = P + SI \]
where \( SI = \frac{P \cdot R \cdot T}{100} \).

For three years:
\[ A = P + \frac{P \cdot R \cdot 3}{100} \]
Setting \( A = 19500 \):
\[ 19500 = P + \frac{3P \cdot R}{100} \]

This equation does not directly help us find \( R \) without knowing \( P \). Thus, we will hold this equation for later use.

### Step 3: Analyze Statement III
Statement III states that the simple interest acquired in two years on the same amount at the same rate of interest is Rs. 3,000.

Using the formula for simple interest:
\[ SI = \frac{P \cdot R \cdot T}{100} \]
For two years:
\[ 3000 = \frac{P \cdot R \cdot 2}{100} \]
Rearranging gives us:
\[ P \cdot R = 3000 \cdot \frac{100}{2} \]
\[ P \cdot R = 150000 \quad \text{(Equation 2)} \]

### Step 4: Solve Equations
Now we have two equations:
1. \( P \cdot R^2 = 10000 \) (from Statement I)
2. \( P \cdot R = 150000 \) (from Statement III)

From Equation 2, we can express \( P \) in terms of \( R \):
\[ P = \frac{150000}{R} \]

Substituting this into Equation 1:
\[ \frac{150000}{R} \cdot R^2 = 10000 \]
\[ 150000R = 10000 \]
\[ R = \frac{10000}{150000} = \frac{1}{15} \]

### Step 5: Find Rate of Interest
To convert \( R \) into percentage:
\[ R = \frac{1}{15} \times 100 = \frac{100}{15} = \frac{20}{3} \]

Thus, the rate of interest per annum (PCPA) is:
\[ \frac{20}{3} \% \]

### Conclusion
The required answer for the rate of interest per annum is \( \frac{20}{3} \% \).