Six years ago, the ratio of the ages of P and Q was 8:3. Sixteen years from now, P's age will be 1.75 times that of Q. Find the ratio of ages of P to Q six years from now.

The correct answer is 2:1.

Step 1: Define variables for the present ages.
Let the present age of P be $A_{\text{P}}$ years and the present age of Q be $A_{\text{Q}}$ years.

Step 2: Set up the equation for their ages six years ago.
Six years ago, P's age was $A_{\text{P}}-6$ and Q's age was $A_{\text{Q}}-6$.
The given ratio of their ages six years ago was 8:3:

$\frac{A_{\text{P}}-6}{A_{\text{Q}}-6}=\frac{8}{3}$

Cross-multiplying to simplify the equation:
$3(A_{\text{P}}-6)=8(A_{\text{Q}}-6)$
$3A_{\text{P}}-18=8A_{\text{Q}}-48$
$3A_{\text{P}}=8A_{\text{Q}}-30$
$A_{\text{P}}=\frac{8A_{\text{Q}}-30}{3}$ (Equation 1)

Step 3: Set up the equation for their ages sixteen years from now.
Sixteen years from now, P's age will be $A_{\text{P}}+16$ and Q's age will be $A_{\text{Q}}+16$.
We are given that P's age will be 1.75 times Q's age.
Note that $1.75=\frac{7}{4}$.

$A_{\text{P}}+16=1.75(A_{\text{Q}}+16)$

$A_{\text{P}}+16=\frac{7}{4}(A_{\text{Q}}+16)$

Multiply both sides by 4:
$4(A_{\text{P}}+16)=7(A_{\text{Q}}+16)$
$4A_{\text{P}}+64=7A_{\text{Q}}+112$
$4A_{\text{P}}-7A_{\text{Q}}=48$ (Equation 2)

Step 4: Solve for the present ages.
Substitute Equation 1 into Equation 2:

$4\left(\frac{8A_{\text{Q}}-30}{3}\right)-7A_{\text{Q}}=48$

Multiply the entire equation by 3 to clear the fraction:
$4(8A_{\text{Q}}-30)-21A_{\text{Q}}=144$
$32A_{\text{Q}}-120-21A_{\text{Q}}=144$
$11A_{\text{Q}}-120=144$
$11A_{\text{Q}}=264$
$A_{\text{Q}}=24$

Now, substitute $A_{\text{Q}}=24$ back into Equation 1 to find $A_{\text{P}}$:
$A_{\text{P}}=\frac{8\left(24\right)-30}{3}=\frac{192-30}{3}=\frac{162}{3}=54$

So, the present age of P is 54 years, and the present age of Q is 24 years.

Step 5: Calculate the ratio of their ages six years from now.
Six years from now:
P's age will be $54+6=60$
Q's age will be $24+6=30$

The ratio of their ages six years from now is:

$\text{Ratio}=\frac{60}{30}=\frac{2}{1}=\mathrm{2:1}$