Ratio of age of P 2 years ago to age of R 2 years hence is 1 : 2 and Q's present age is 25% more than P's present age. If average of present age of P& R is 39 years, then find difference between P's age 5 years hence and R's present age.

Options

A. A)12 years
B. B)17 years
C. C)21 years
D. D)15 years

Answer

To solve the problem step by step, we will follow the information given in the question and derive the necessary equations to find the required ages.

### Step 1: Set up the ratio of ages
We know that the ratio of the age of P two years ago to the age of R two years hence is 1:2. Let's denote P's present age as \( P \) and R's present age as \( R \).

From the information given:
- P's age two years ago = \( P - 2 \)
- R's age two years hence = \( R + 2 \)

According to the ratio:
\[ \frac{P - 2}{R + 2} = \frac{1}{2} \]

### Step 2: Cross-multiply to find a relationship
Cross-multiplying gives us:
\[ 2(P - 2) = 1(R + 2) \]
Expanding this:
\[ 2P - 4 = R + 2 \]
Rearranging gives us:
\[ R = 2P - 6 \quad \text{(Equation 1)} \]

### Step 3: Use the average age information
We are told that the average of the present ages of P and R is 39 years. Therefore:
\[ \frac{P + R}{2} = 39 \]
Multiplying both sides by 2:
\[ P + R = 78 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
Now, we can substitute the expression for \( R \) from Equation 1 into Equation 2:
\[ P + (2P - 6) = 78 \]
Combining like terms:
\[ 3P - 6 = 78 \]
Adding 6 to both sides:
\[ 3P = 84 \]
Dividing by 3:
\[ P = 28 \quad \text{(P's present age)} \]

### Step 5: Find R's present age
Now we can find R's present age using Equation 1:
\[ R = 2P - 6 = 2(28) - 6 = 56 - 6 = 50 \quad \text{(R's present age)} \]

### Step 6: Calculate P's age 5 years hence
P's age 5 years hence will be:
\[ P + 5 = 28 + 5 = 33 \]

### Step 7: Find the difference between P's age 5 years hence and R's present age
Now, we need to find the difference between P's age 5 years hence and R's present age:
\[ \text{Difference} = R - (P + 5) = 50 - 33 = 17 \]

Thus, the difference between P's age 5 years hence and R's present age is **17 years**.

### Final Answer
The difference between P's age 5 years hence and R's present age is **17 years**.

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