To solve the problem step by step, we will use the information given about the ratios and the conditions provided.
### Step 1: Set Up the Ratios
Let the two numbers be represented as \( P \) and \( Q \). The initial ratio can be expressed as:
\[
\frac{P}{Q} = \frac{P}{Q}
\]
### Step 2: First Condition
According to the problem, when 1 is added to both the numerator and the denominator, the ratio changes to \( \frac{R}{S} \):
\[
\frac{P + 1}{Q + 1} = \frac{R}{S}
\]
### Step 3: Second Condition
Again, when 1 is added to both the numerator and the denominator, it becomes \( \frac{1}{2} \):
\[
\frac{P + 1}{Q + 1} = \frac{1}{2}
\]
### Step 4: Cross-Multiply to Form an Equation
From the second condition, we can cross-multiply to eliminate the fraction:
\[
2(P + 1) = 1(Q + 1)
\]
Expanding this gives:
\[
2P + 2 = Q + 1
\]
Rearranging this, we get:
\[
2P - Q = -1 \quad \text{(Equation 1)}
\]
### Step 5: Express Q in Terms of P
From Equation 1, we can express \( Q \) in terms of \( P \):
\[
Q = 2P + 1
\]
### Step 6: Substitute into the First Condition
Now, we substitute \( Q \) back into the first condition:
\[
\frac{P + 1}{(2P + 1) + 1} = \frac{R}{S}
\]
This simplifies to:
\[
\frac{P + 1}{2P + 2} = \frac{R}{S}
\]
### Step 7: Find Values of P and Q
To find suitable values for \( P \) and \( Q \), we will check the values of \( P + Q \) given in the options. We will try different values for \( P \) and \( Q \) to see which satisfies both conditions.
1. **Option A: \( P + Q = 3 \)**
- Possible pairs: \( (1, 2) \) or \( (2, 1) \)
- Check \( (1, 2) \):
\[
\frac{1 + 1}{2 + 1} = \frac{2}{3} \quad \text{(not } \frac{1}{2}\text{)}
\]
- Check \( (2, 1) \):
\[
\frac{2 + 1}{1 + 1} = \frac{3}{2} \quad \text{(not } \frac{1}{2}\text{)}
\]
2. **Option B: \( P + Q = 4 \)**
- Possible pairs: \( (1, 3) \) or \( (2, 2) \) or \( (3, 1) \)
- Check \( (1, 3) \):
\[
\frac{1 + 1}{3 + 1} = \frac{2}{4} = \frac{1}{2} \quad \text{(valid)}
\]
3. **Option C: \( P + Q = 5 \)**
- Possible pairs: \( (2, 3) \) or \( (3, 2) \) or \( (4, 1) \)
- Check \( (2, 3) \):
\[
\frac{2 + 1}{3 + 1} = \frac{3}{4} \quad \text{(not } \frac{1}{2}\text{)}
\]
### Conclusion
The only valid option that satisfies both conditions is when \( P + Q = 4 \). Therefore, the sum of \( P \) and \( Q \) is:
\[
\boxed{4}
\]
