Ages Problems
Age problems involve tracking ratios that change across past, present, and future timelines while the absolute difference between ages stays the same.
Fundamental Principles
Timeline Invariance
The mathematical fact that the age difference between two people never changes, no matter how many years pass into the past or future.
Essential Formulation Tips
- When adding or subtracting years to shift a timeline, remember to apply the change to both people's ages.
- Clearly label your algebraic variables to separate past ratios, present ratios, and future ratios.
Shortcut Execution Techniques
- If the ratio of ages changes from 2 : 3 to 3 : 4 over a span of 5 years, notice that both ratio parts increase by exactly 1 unit. This means 1 ratio unit equals 5 years.
Contextual Inquiries (FAQs)
Q: Why can't we just cross-multiply ratios from different years directly?
A: Because adding a fixed number of years changes the value of the ratio itself, even though the absolute difference between the ages stays the same.
Example Breakdown: Calculating a Present Age from a Past Ratio
Classic age timeline problem.Let the current ages be 7x and 3x.
Set up an equation for their ages 4 years ago: (7x - 4) / (3x - 4) = 3 / 1.
Cross-multiply to solve: 1 * (7x - 4) = 3 * (3x - 4) -> 7x - 4 = 9x - 12.
Rearrange the terms: 2x = 8 -> x = 4.
Calculate the mother's current age: 7x = 7 * 4 = 28 years old.
Age Ratio Timelines
Practice solving age problems across past, present, and future timelines.
Q1. The ratio of two brothers' ages is 4 : 5. If the sum of their ages is 36, how old is the younger brother?