Seven Times P Age Relation

This problem asks us to find the current age of Q based on relationships involving the ages of P and Q. We will use algebra to set up and solve equations representing these relationships.

Let $P$ represent the present age of person P, and let $Q$ represent the present age of person Q.

First Condition: Present Ages

The problem states: "Seven times the present age of P exceeds the present age of Q by 46 years." This translates directly into an algebraic equation:

$7P = Q + 46$

We can rewrite this equation as:

$7P - Q = 46 \quad (1)$

Ages After 8 Years Setup

The problem also describes a situation 8 years from now. In 8 years:

• P's age will be $P + 8$.
• Q's age will be $Q + 8$.

The second condition is: "six times Q's age will be 24 years less than six times P's age at that time." This gives us the equation:

$6(Q + 8) = 6(P + 8) - 24$

Let's simplify this equation:

$6Q + 48 = 6P + 48 - 24$

$6Q + 48 = 6P + 24$

Subtracting 48 from both sides:

$6Q = 6P - 24$

Dividing every term by 6 simplifies the equation to:

$Q = P - 4 \quad (2)$

Solving P and Q Age Equations

We now have a system of two linear equations:

1. $7P - Q = 46$
2. $Q = P - 4$

We can use the substitution method. From equation (2), we can express $P$ in terms of $Q$ as $P = Q + 4$.

Substitute $P = Q + 4$ into equation (1):

$7(Q + 4) - Q = 46$

Distribute the 7:

$7Q + 28 - Q = 46$

Combine the terms involving $Q$:

$6Q + 28 = 46$

Isolate the $Q$ term by subtracting 28 from both sides:

$6Q = 46 - 28$

$6Q = 18$

Solve for $Q$ by dividing by 6:

$Q = \frac{18}{6}$

$Q = 3$

Age Calculation Verification

To ensure our answer is correct, let's verify it using the original conditions.

If $Q = 3$, then using $P = Q + 4$, we find $P = 3 + 4 = 7$. So, P is 7 years old and Q is 3 years old.

Verifying Condition 1:

Is $7 \times P$ equal to $Q + 46$? $7 \times 7 = 49$ $3 + 46 = 49$ Yes, $49 = 49$. Condition 1 holds true.

Verifying Condition 2:

After 8 years, P will be $7 + 8 = 15$ and Q will be $3 + 8 = 11$.

Is $6 \times (Q \text{'s future age})$ equal to $6 \times (P \text{'s future age}) - 24$? $6 \times 11 = 66$ $6 \times 15 - 24 = 90 - 24 = 66$ Yes, $66 = 66$. Condition 2 holds true.

Q's Present Age Found

Since both conditions are satisfied, the calculated present age of Q is correct.

The present age of Q is 3 years.