Understanding the Age Problem Details

This problem involves finding the current age of a person named Q. We are given two pieces of information that relate the present ages of two people, P and Q, and their ages after a specific time period (10 years). To solve this, we need to translate these statements into mathematical equations.

Setting Up Algebraic Equations for Ages

Let's denote the present age of P as '$P$' years and the present age of Q as '$Q$' years. We can now translate the given conditions into equations:

• Condition 1: "Six times the present age of P exceeds the present age of Q by 11 years." This means that $6 \times P$ is 11 more than $Q$. The equation is: $6P = Q + 11$ Rearranging this gives us our first equation: $6P - Q = 11$ (Equation 1)
• Condition 2: "After 10 years, three times Q's age will be 27 years less than six times P's age at that time." First, let's determine their ages after 10 years: P's age after 10 years = $P + 10$ Q's age after 10 years = $Q + 10$ Now, let's translate the condition: $3 \times (Q + 10)$ is equal to $6 \times (P + 10)$ minus 27. The equation is: $3(Q + 10) = 6(P + 10) - 27$ Let's simplify this equation: $3Q + 30 = 6P + 60 - 27$ $3Q + 30 = 6P + 33$ Rearranging the terms to match the format of Equation 1: $6P - 3Q = 30 - 33$ $6P - 3Q = -3$ (Equation 2)

Solving the System of Equations for Q's Age

We now have a system of two linear equations with two variables ($P$ and $Q$): 1. $6P - Q = 11$ 2. $6P - 3Q = -3$ We can solve this system using the elimination method. Notice that both equations have a '$6P$' term. If we subtract Equation 2 from Equation 1, the '$P$' term will be eliminated.

Subtracting Equation 2 from Equation 1: $ (6P - Q) - (6P - 3Q) = 11 - (-3) $ $ 6P - Q - 6P + 3Q = 11 + 3 $ Combine like terms: $ (-Q + 3Q) + (6P - 6P) = 14 $ $ 2Q = 14 $ Now, solve for $Q$ by dividing both sides by 2: $ Q = \frac{14}{2} $ $ Q = 7 $ So, the present age of Q is 7 years.

Calculating P's Age for Verification

Although the question only asks for Q's age, we can find P's age to verify our answer using Equation 1: $6P - Q = 11$. Substitute $Q = 7$: $ 6P - 7 = 11 $ Add 7 to both sides: $ 6P = 11 + 7 $ $ 6P = 18 $ Divide by 6: $ P = \frac{18}{6} $ $ P = 3 $ So, P's present age is 3 years.

Final Verification of Conditions

Let's check if these ages ($P=3$, $Q=7$) satisfy the conditions given in the problem, especially the second condition after 10 years.

• Condition 1 Check: $6P = 6 \times 3 = 18$. $Q + 11 = 7 + 11 = 18$. This condition holds true ($18 = 18$).
• Condition 2 Check (After 10 years): P's age = $3 + 10 = 13$ years. Q's age = $7 + 10 = 17$ years. Three times Q's age = $3 \times 17 = 51$. Six times P's age = $6 \times 13 = 78$. Is $3 \times Q$'s age (51) equal to $6 \times P$'s age (78) minus 27? $78 - 27 = 51$. Yes, $51 = 51$. This condition also holds true.

Both conditions are satisfied with $P=3$ and $Q=7$.

Conclusion: Q's Present Age

Based on the calculations and verification, the present age of Q is 7 years.