Introduction
Age problems connect a person's age at different points in time (past, present, future) through ratios or
differences. The fastest approach is to assign the current ratio a variable multiplier and solve with a single
linear equation.
Standard Setup
If present ages are in ratio a:b, let them be "ax" and "bx." Use the given condition (after n years, before n
years, sum of ages, etc.) to form one equation in x, then solve.
Q. The present ages of A and B are in the ratio 3:4. After 5 years, the ratio becomes 4:5. Find their present ages.
Let present ages = 3x and 4x. After 5 years: (3x+5)/(4x+5) = 4/5.
Cross-multiply: 5(3x+5) = 4(4x+5) → 15x+25 = 16x+20 → x = 5.
Present ages = 3×5=15 years and 4×5=20 years.
Q. Five years ago, the ratio of ages of P and Q was 3:5. Ten years from now, the ratio will be 5:7. Find their present ages.
Let ages 5 years ago = 3x and 5x. Present ages = 3x+5 and 5x+5. Ten years from now = 3x+15 and 5x+15.
(3x+15)/(5x+15) = 5/7 → 7(3x+15) = 5(5x+15) → 21x+105 = 25x+75 → 4x = 30 → x = 7.5.
Present ages = 3(7.5)+5=27.5 years and 5(7.5)+5=42.5 years.
⚠️ Key Insight: Always add or subtract the SAME number of years to both people's ages when
moving to a "years ago" or "years from now" scenario — a frequent error is adjusting only one person's age.
✅ Practice Focus: Ratio-based present age setup · Past/future ratio-change equations · Sum and
difference of ages combined with ratio conditions.