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Arithmetic (Beginner) · Quick Reference
Age Problems Cheat Sheet
In one line: age problems use the fact that everyone's age increases by the same amount each year, letting us set up simple equations to find unknown ages.
Key Ideas
1Present vs. Past/Future Age. If present age is x, age n years ago was (x - n), and age n years from now will be (x + n).
2Setting Up Equations. Translate word relationships ('twice as old', 'sum of ages') directly into algebraic equations using a variable for one age.
3Ratio of Ages. When ages are given as a ratio, represent them using a common multiplier (e.g., 3x and 5x).
4Age Difference is Constant. The difference between two people's ages never changes over time, even though both ages increase.
Worked Examples
A father is 3 times as old as his son. In 5 years, he will be 2.5 times as old. Find their present ages.
Son = 15, Father = 45
The sum of a mother's and daughter's ages is 50. The mother is 20 years older. Find both ages.
Daughter = 15, Mother = 35
The ratio of two friends' ages is 4:5, and the older is 20. Find the younger's age.
16
Practice Yourself
A is twice as old as B. Their ages sum to 30. Find both ages.
A=20, B=10
5 years ago, a person was 20. What is their age now?
25
In 10 years, a person will be 40. What is their age now?
30
Ages are in ratio 2:3, and the younger is 14. Find the older's age.
21