We need to find the present age of Q based on the given conditions involving the ages of P and Q.

Age Problem Setup

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

Equation Formulation

Translate the conditions into algebraic equations:

• "4 times the present age of P exceeds the present age of Q by 19 years": $4P = Q + 19 \quad \cdots (1)$
• "After 5 years": P's age will be $(P+5)$ and Q's age will be $(Q+5)$.
• "3 times the age of Q will be 27 years less than 9 the age of P": $3(Q+5) = 9(P+5) - 27 \quad \cdots (2)$

Solving the Equations

Simplify Equation (2):

• $3Q + 15 = 9P + 45 - 27$
• $3Q + 15 = 9P + 18$
• $3Q - 9P = 18 - 15$
• $3Q - 9P = 3$
• Divide the entire equation by 3: $Q - 3P = 1 \quad \cdots (3)$

Now, use Equation (1) to express $Q$ in terms of $P$:

• From Equation (1): $Q = 4P - 19$.

Substitute this expression for $Q$ into Equation (3):

• $(4P - 19) - 3P = 1$
• $P - 19 = 1$
• $P = 1 + 19$
• $P = 20$

Finding Q's Age

Substitute the value of $P = 20$ back into the expression for $Q$ (derived from Equation 1):

• $Q = 4P - 19$
• $Q = 4(20) - 19$
• $Q = 80 - 19$
• $Q = 61$

Therefore, the present age of Q is 61 years.