Four years from now, the ratio of the ages of P and Q will be 3 : 5, respectively. The ratio of the present ages of R and Q is 5 : 6, respectively. If the sum of the ages of P and R six years from now is 62 years, what is the sum of the present ages of P and Q?

Correct Option: 56 years

Let us break down the given problem step-by-step to find the sum of the present ages of P and Q.

Step 1: Define variables for the present ages
Let the present ages of P, Q, and R be denoted by $P$, $Q$, and $R$ years respectively.

Step 2: Formulate equations from the given conditions

Condition 1: Four years from now, the ratio of the ages of P and Q will be 3 : 5.
Four years from now, P's age will be $P+4$ and Q's age will be $Q+4$.

$\frac{P+4}{Q+4}=\frac{3}{5}$

Cross-multiplying to simplify:
$5(P+4)=3(Q+4)$
$5P+20=3Q+12$
$5P-3Q=-8$ --- (Equation 1)

Condition 2: The ratio of the present ages of R and Q is 5 : 6.
$\frac{R}{Q}=\frac{5}{6}$
Therefore, we can express $R$ in terms of $Q$:

$R=\frac{5Q}{6}$ --- (Equation 2)

Condition 3: The sum of the ages of P and R six years from now is 62 years.
Six years from now, P's age will be $P+6$ and R's age will be $R+6$.

$(P+6)+(R+6)=62$

$P+R+12=62$
$P+R=50$ --- (Equation 3)

Step 3: Solve for P and Q

Substitute Equation 2 into Equation 3:

$P+\frac{5Q}{6}=50$

Multiplying the entire equation by 6:
$6P+5Q=300$
$5Q=300-6P$
$Q=\frac{300-6P}{5}$ --- (Equation 4)

Now, substitute Equation 4 into Equation 1:

$5P-3\left(\frac{300-6P}{5}\right)=-8$

Multiply the entire equation by 5 to clear the fraction:
$25P-3(300-6P)=-40$
$25P-900+18P=-40$
$43P=900-40$
$43P=860$

$P=\frac{860}{43}=20\text{years}$

Now, substitute $P=20$ into Equation 4 to find $Q$:

$Q=\frac{300-6\left(20\right)}{5}=\frac{300-120}{5}=\frac{180}{5}=36\text{years}$

Step 4: Find the sum of the present ages of P and Q

$\text{Sum}=P+Q=20+36=56\text{years}$

Thus, the sum of the present ages of P and Q is 56 years.