Twice the sum of the ages of a father and his son is eight times the son's age. If the average age of the father and the son is 30 years, what is the father's present age?

Introduction / Context: This question combines a relationship between the sum of two ages and a multiple of one age with information about their average age. By setting up equations for the father and son's ages, we can solve for the father's present age.

Given Data / Assumptions:

- Let the father's present age be F years and the son's present age be S years. - Twice the sum of their ages is eight times the son's age: 2(F + S) = 8S. - The average age of the father and son is 30 years, so (F + S) / 2 = 30. - We must find the father's age F.

Concept / Approach: We translate the verbal statements into algebraic equations and then solve the system. From the equation involving the average, we can find the sum F + S. Substituting this sum into the first equation allows us to solve for S, and then F is obtained using the sum. This method keeps the calculations straightforward and organized.

Step-by-Step Solution: Step 1: From the average condition, (F + S) / 2 = 30. Step 2: Multiply both sides by 2 to get F + S = 60. Step 3: From the first condition, 2(F + S) = 8S. Step 4: Substitute F + S = 60 into 2(F + S) = 8S to get 2 × 60 = 8S. Step 5: This simplifies to 120 = 8S, so S = 120 / 8 = 15 years. Step 6: Use F + S = 60 again to find F = 60 − S = 60 − 15 = 45 years.

Verification / Alternative check: Check both conditions with F = 45 and S = 15. The average age is (45 + 15) / 2 = 60 / 2 = 30 years, which matches the given average. The sum of the ages is 60, so twice the sum is 2 × 60 = 120. Eight times the son's age is 8 × 15 = 120, which matches the other condition. Both statements are satisfied, confirming that F = 45 years is correct.

Why Other Options Are Wrong: Ages such as 36, 38, 42, or 48 years for the father will not simultaneously satisfy both the average condition and the equation 2(F + S) = 8S when the son's age is calculated. Only 45 years leads to a consistent pair of ages (45 and 15) fulfilling both requirements.

Common Pitfalls: Some students may misinterpret "twice the sum" as "the sum is twice" or incorrectly form the relation as F + S = 8S. Others may forget to use the average equation to find F + S. The key is to write each condition carefully and solve the system step by step without skipping algebraic steps.

Final Answer: The father's present age is 45 years.