Two Years Ago a Man was Five Times

This is another age-related algebra problem, likely continuing with additional information about future ages and asking to find current ages. The complete problem might read: "Two years ago, a man was five times as old as his son. [Additional condition about future or present ages]. Find their current ages." Without the complete problem, I'll demonstrate the systematic approach to solving such multi-step age problems.

Let's assume the complete problem is: "Two years ago, a man was five times as old as his son. In two years, he will be three times as old as his son. Find their present ages." Solution approach: (1) Define variables—let son's present age = x years, father's present age = y years; (2) Translate first condition: Two years ago, son's age was (x - 2) and father's age was (y - 2), with father being five times son's age: y - 2 = 5(x - 2), which simplifies to y - 2 = 5x - 10, giving y = 5x - 8; (3) Translate second condition: In two years, son's age will be (x + 2) and father's age will be (y + 2), with father being three times son's age: y + 2 = 3(x + 2), which simplifies to y + 2 = 3x + 6, giving y = 3x + 4; (4) Solve simultaneous equations: Since y = 5x - 8 and y = 3x + 4, we have 5x - 8 = 3x + 4, solving gives 2x = 12, so x = 6; (5) Find both ages: Son's present age = 6 years, Father's present age = 5(6) - 8 = 22 years (or using second equation: 3(6) + 4 = 22); (6) Verify: Two years ago, son was 4 and father was 20 (20 = 5 × 4 ✓); In two years, son will be 8 and father will be 24 (24 = 3 × 8 ✓). These problems develop: algebraic translation skills (converting verbal statements to equations), simultaneous equation solving, logical reasoning (understanding age relationships), and verification habits (checking answers make sense). Common variations include: ages n years ago/hence, ratios of ages, sums or differences of ages, and problems involving multiple family members. Keys to success include: careful reading (identify all relationships), systematic variable assignment, accurate equation formation, proper algebraic manipulation, and always checking solutions against original problem conditions. These mathematical exercises, while abstract, build quantitative reasoning applicable beyond mathematics—understanding relationships, translating between verbal and mathematical representations, systematic problem-solving, and verification—skills valuable across academic, professional, and everyday contexts requiring logical thinking and analytical problem-solving.