🎂 Ages

Day 1 | ⏱️ Part of 4-Hour Quant Block | TCS Interview Prep

📌 Subtopics

• Problems based on present age, past age, future age
• Age ratios at different times
• Ages and fractions
• Ages with multiple people
• Word problems → Linear equations approach

📖 Comprehensive Notes

1. Present, Past, and Future Ages

The key to solving age problems is setting up algebraic expressions relative to a reference point (usually the present age).

• If the present age is $x$ years:
  • Age $n$ years ago $= x - n$
  • Age $n$ years from now (hence) $= x + n$

Example: A man is currently $3$ times as old as his son. In $12$ years, he will be twice as old as his son. Find their present ages. Let son’s present age $= s$. Man’s present age $= 3s$. In $12$ years: Son’s age $= s + 12$ Man’s age $= 3s + 12$ Equation: $3s + 12 = 2(s + 12) \implies 3s + 12 = 2s + 24 \implies s = 12$. So, son’s age $= 12$ years, father’s age $= 36$ years.

2. Age Ratios at Different Times

Ratios of ages change over time, but the difference between the ages of two people remains constant throughout their lives.

• If A’s age is $30$ and B’s age is $20$, the difference is $10$. In $50$ years, A will be $80$ and B will be $70$; the difference is still $10$.

Example: The ratio of ages of A and B is $4:5$. Six years ago, the ratio of their ages was $3:4$. Find their present ages. Let A’s age $= 4x$, B’s age $= 5x$. Six years ago: $\frac{4x - 6}{5x - 6} = \frac{3}{4}$ Cross-multiplying: $4(4x - 6) = 3(5x - 6) \implies 16x - 24 = 15x - 18 \implies x = 6$. Present ages: A $= 24$ years, B $= 30$ years.

3. Ages and Fractions

Sometimes relationships are expressed as fractions (e.g., “A’s age is $\frac{2}{5}$ of B’s age”).

• Set $B = x$, then $A = \frac{2}{5}x$. Or, more conveniently to avoid fractions, let $B = 5x$ and $A = 2x$.

Example: Raj’s age is $\frac{1}{6}$ of his father’s age. If the difference between their ages is $35$ years, find Raj’s age. Let Father’s age $= 6x$, Raj’s age $= x$. Difference: $6x - x = 35 \implies 5x = 35 \implies x = 7$. Raj’s age $= 7$ years.

4. Ages with Multiple People

When dealing with three or more people, represent all their ages in terms of a single variable if possible to make the equations simpler.

Example: The sum of the ages of $5$ children born at intervals of $3$ years each is $50$ years. Find the age of the youngest child. Let the youngest child’s age $= x$. Ages of the children: $x, x+3, x+6, x+9, x+12$. Sum $= 5x + 30 = 50 \implies 5x = 20 \implies x = 4$. Youngest child is $4$ years old.

5. Word Problems to Linear Equations

1. Clearly define what the variables represent (e.g., $M$ for Mother, $D$ for Daughter).
2. Carefully translate words to mathematical operations:
   • “Times” $\rightarrow$ Multiplication ($\times$)
   • “Exceeds/Older than” $\rightarrow$ Addition ($+$)
   • “Ago/Before” $\rightarrow$ Subtraction ($-$)
   • “Hence/Later/After” $\rightarrow$ Addition ($+$)
3. Solve the resulting system of linear equations.

🌐 Learning Resources

✅ Practice Links

📝 Key Approach

1. Let current age = x
2. "n years ago" → (x - n)
3. "n years later" → (x + n)
4. Form equation based on the condition
5. Solve for x

Tip: Always cross-check by substituting your answer back into the original problem statement.

🔗 Related Topics

• Equations — Age problems are solved using linear equations
• Ratios and Proportion — Age ratio problems at different times
• Averages — Average age of a group
• Blood Relations — Age clues in family relation puzzles
• Number System HCF LCM — Number properties in age word problems