Sum of present ages of P and Q is 41. Age of P 2 year hence is equal to age of R, 1 year ago. Age of P, 4 year hence is equal to age of Q 1 year ago and ratio of present age of P and S is 3 : 4. Find the difference of age of R and S.

Options

A. A) 2
B. B) 3
C. C) 4
D. D) 5

Answer

To solve the problem step by step, we will define the variables for the ages of P, Q, R, and S, and then use the information given in the question to set up equations.

### Step 1: Define the variables
Let:
- Age of P = p
- Age of Q = q
- Age of R = r
- Age of S = s

### Step 2: Set up the equations based on the information given
1. The sum of the present ages of P and Q is 41:
\[ p + q = 41 \quad \text{(Equation 1)} \]

2. The age of P two years hence (in the future) is equal to the age of R one year ago (in the past):
\[ p + 2 = r - 1 \quad \Rightarrow \quad r = p + 3 \quad \text{(Equation 2)} \]

3. The age of P four years hence is equal to the age of Q one year ago:
\[ p + 4 = q - 1 \quad \Rightarrow \quad q = p + 5 \quad \text{(Equation 3)} \]

4. The ratio of the present ages of P and S is 3:4:
\[ \frac{p}{s} = \frac{3}{4} \quad \Rightarrow \quad s = \frac{4}{3}p \quad \text{(Equation 4)} \]

### Step 3: Substitute Equation 3 into Equation 1
From Equation 3, we have:
\[ q = p + 5 \]
Substituting this into Equation 1:
\[ p + (p + 5) = 41 \]
\[ 2p + 5 = 41 \]
\[ 2p = 41 - 5 \]
\[ 2p = 36 \]
\[ p = 18 \]

### Step 4: Find the age of Q using the value of P
Using the value of \( p \) in Equation 3:
\[ q = p + 5 = 18 + 5 = 23 \]

### Step 5: Find the age of R using the value of P
Using the value of \( p \) in Equation 2:
\[ r = p + 3 = 18 + 3 = 21 \]

### Step 6: Find the age of S using the value of P
Using the value of \( p \) in Equation 4:
\[ s = \frac{4}{3}p = \frac{4}{3} \times 18 = 24 \]

### Step 7: Find the difference between the ages of R and S
Now we can find the difference between the ages of R and S:
\[ \text{Difference} = s - r = 24 - 21 = 3 \]

### Final Answer
The difference between the ages of R and S is **3 years**.

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