Problems on Ages
Scenario
Every coding round has that puzzle: 'A father is twice as old as his son...'
How do you translate a family tree spanning past, present, and future into a simple linear equation?
Mental model
The age difference between two people never changes. Only the ratio of their ages changes over time.
Age problems translate past, present, and future timelines into linear equations using a single variable for the present age.
Formula
Let present age = x. Age n years ago = x - n. Age n years from now = x + n. If present ratio A:B = m:n, then A = mk and B = nk for some k. Equations from different time points -> solve for k or x. Constant difference: A - B = (A-n) - (B-n), always the same.
Always anchor your variable to the Present Age to keep equations stable. Set up one equation per given condition. If two people's present ratio and a future or past ratio are both given, use the constant difference to equate them without needing simultaneous equations.
Pattern discovery
Track the ages of two people as they travel through time together.
- Ages 10 & 15 -> Diff = 5
- Ages 20 & 25 (after 10 years) -> Diff = 5
- Ages 50 & 55 (after 40 years) -> Diff = 5
While ratios of ages constantly change over time, what metric remains absolutely constant?
Rule: The age difference between two people never changes. This constant difference is the key to solving ratio-based age problems.
Worked example
Problem: The ratio of the ages of Aman and Bimal is 4:7. After 10 years, the ratio will be 3:4. Find their present ages.
- Set up: Let Aman's present age = 4k, Bimal's = 7k.
- After 10 years: (4k + 10) / (7k + 10) = 3/4.
- Cross-multiply: 4(4k + 10) = 3(7k + 10) -> 16k + 40 = 21k + 30.
- 5k = 10 -> k = 2.
- Present ages: Aman = 4 x 2 = 8, Bimal = 7 x 2 = 14.
- Sanity-check: After 10 years, 18:24 = 3:4. Correct.
Answer: Aman is 8 years old, Bimal is 14 years old.
Shortcuts
- The Constant Difference Trick: Since age difference never changes, if you have two ratios at two different times, multiply the ratios by factors to make their difference equal, then each unit equals the years elapsed. (Past ratio 1:2 (diff 1), Future ratio 3:4 (diff 1). Difference is already equal. If the time gap is 10 years, 2 units of ratio = 10 years, 1 unit = 5 years.)
- Checking Multiples: If a person's present age ratio part is 5, their actual age is likely a multiple of 5. Use this to eliminate MCQ options rapidly. (If A:B is 5:3 and A's age is a multiple of 5, from options 18, 21, 25, 28 - pick 25.)
Common mistakes
- Adding Years to Ratios directly: If the present ratio is 2:3, the ages after 5 years are NOT 7:8. The ages are 2x+5 and 3x+5, which simplify to a different ratio.
- Wrong Baseline Variable: Setting 'x' as the past age, solving for 'x', and selecting 'x' as the answer when the question asked for the present age.
Glossary
- variable
- A letter like x used in math to represent an unknown number.
- timelines
- Different points in time, like the past, present, and future.
- equate
- To set two things as equal to each other in an equation.
Recall questions
- If A is 10 years older than B today, how much older will A be than B in 15 years?
- Present ages of A and B are in ratio 4:5. If A is 20, what is B's age?
- Ten years ago, P was 20. What will be his age 5 years from now?
Questions & answers
A father said to his son, 'I was as old as you are at present at the time of your birth.' If the father's age is 38 years now, the son's age five years back was:
14 years
Approach: Let son's present age be x. Father's age at son's birth was (38 - x). Given: 38 - x = x -> 2x = 38 -> x = 19. Son's age 5 years back = 19 - 5 = 14.
The sum of ages of 5 children born at intervals of 3 years each is 50 years. What is the age of the youngest child?
4 years
Approach: Let youngest be x. Ages: x, x+3, x+6, x+9, x+12. Sum = 5x + 30 = 50. 5x = 20 -> x = 4.