Age Problem Analysis

This question requires us to determine the present age of an individual named Q, given specific relationships between the present ages of P and Q, and their ages after 9 years. We can solve this using algebra by setting up and solving equations.

Formulating Equations from the Problem Statement

First, let's define variables for the unknown ages:

• Let $P$ be the present age of P (in years).
• Let $Q$ be the present age of Q (in years).

Translating the First Condition

The problem states: "Two times the present age of P exceeds the present age of Q by 15 years". This translates to the algebraic equation:

$2P = Q + 15 \quad (Equation \, 1)$

Translating the Second Condition

We need to consider their ages after 9 years:

• P's age after 9 years will be $P + 9$.
• Q's age after 9 years will be $Q + 9$.

The problem states: "After 9 years, twice Q's age will be 14 years less than twice P's age at that time". This gives us the equation:

$2(Q + 9) = 2(P + 9) - 14 \quad (Equation \, 2)$

Solving the System of Equations

Our goal is to find the value of $Q$. We have a system of two linear equations with two variables:

1. $2P = Q + 15$
2. $2(Q + 9) = 2(P + 9) - 14$

Step 1: Simplify Equation 2

Let's simplify the second equation:

$2Q + 18 = 2P + 18 - 14$

Combine the constants on the right side:

$2Q + 18 = 2P + 4$

Rearrange the terms to make it easier to substitute or solve:

$2Q = 2P + 4 - 18$

$2Q = 2P - 14 \quad (Simplified \, Equation \, 2)$

Step 2: Substitute Equation 1 into Simplified Equation 2

From Equation 1, we know that $2P = Q + 15$. We can substitute this expression for $2P$ into the simplified Equation 2:

$2Q = (Q + 15) - 14$

Step 3: Solve for Q

Now, we have a single equation with only the variable $Q$. Let's solve it:

$2Q = Q + 15 - 14$

$2Q = Q + 1$

Subtract $Q$ from both sides of the equation:

$2Q - Q = 1$

$Q = 1$

Step 4: Determine Q's Present Age

The value of $Q$ represents the present age of Q. Therefore, Q's present age is 1 year.

Verification (Optional)

To ensure our answer is correct, let's find P's age and check the conditions.

Using Equation 1 with $Q=1$:

$2P = 1 + 15$

$2P = 16$

$P = 8$

So, P is currently 8 years old and Q is 1 year old.

Now, let's check the condition after 9 years:

• P's age = $8 + 9 = 17$
• Q's age = $1 + 9 = 10$

Twice Q's age = $2 \times 10 = 20$.

Twice P's age = $2 \times 17 = 34$.

Is twice Q's age (20) equal to 14 less than twice P's age (34)?

$34 - 14 = 20$

Yes, $20 = 20$. The conditions are met, confirming our answer.

Final Answer

The present age of Q is calculated to be 1 year.