If the difference between the present age of P and Q is three years and the ratio between the age of P and Q after two years will be 5:4, then find the age of P after two years (in-years)?

Options

A. A)15
B. B)13
C. C)18
D. D)16

Answer

To solve the problem step by step, we will use the information given about the ages of P and Q.

### Step 1: Set up the equations based on the information given.
We know that the difference between the present ages of P and Q is 3 years. We can express this as:
\[ P - Q = 3 \]
(Equation 1)

### Step 2: Express the ages of P and Q after 2 years.
After 2 years, the age of P will be \( P + 2 \) and the age of Q will be \( Q + 2 \). According to the problem, the ratio of their ages after 2 years is 5:4. We can express this as:
\[ \frac{P + 2}{Q + 2} = \frac{5}{4} \]
Cross-multiplying gives us:
\[ 4(P + 2) = 5(Q + 2) \]
Expanding this, we get:
\[ 4P + 8 = 5Q + 10 \]
Rearranging gives us:
\[ 4P - 5Q = 2 \]
(Equation 2)

### Step 3: Solve the system of equations.
Now we have two equations:
1. \( P - Q = 3 \) (Equation 1)
2. \( 4P - 5Q = 2 \) (Equation 2)

From Equation 1, we can express \( P \) in terms of \( Q \):
\[ P = Q + 3 \]

Now, substitute \( P \) into Equation 2:
\[ 4(Q + 3) - 5Q = 2 \]
Expanding this gives:
\[ 4Q + 12 - 5Q = 2 \]
Combining like terms results in:
\[ -Q + 12 = 2 \]
Subtracting 12 from both sides gives:
\[ -Q = 2 - 12 \]
\[ -Q = -10 \]
Thus, we find:
\[ Q = 10 \]

### Step 4: Find the age of P.
Now that we have the age of Q, we can find the age of P using Equation 1:
\[ P = Q + 3 = 10 + 3 = 13 \]

### Step 5: Calculate the age of P after 2 years.
To find the age of P after 2 years, we add 2 to P's current age:
\[ P + 2 = 13 + 2 = 15 \]

Thus, the age of P after 2 years is **15 years**.

### Final Answer:
The age of P after two years is **15 years**.

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