Age Problems

Learn to solve word problems involving ages, including present ages, past ages, and future ages.

Intermediate25 minLesson
Printable worksheet

Definition

Age problems are word problems where we use algebra to find the ages of people at different points in time.
The key to solving age problems is:
  1. 1.Define a variable for the unknown age
  2. 2.Translate the relationships into algebraic expressions
  3. 3.Set up an equation based on the given condition
  4. 4.Solve for the variable
Common relationships:
  • "xx years ago" means subtract xx from current age
  • "In xx years" means add xx to current age
  • "Twice as old" means multiply by 22
  • "nn years older" means add nn to the other person's age

Try it now

Lisa is 3 years older than her brother. If her brother is 8 years old, how old is Lisa?

Worked Examples

Maria is 4 years older than her brother Juan. The sum of their ages is 22. How old is each person?
1

Define the variable

Let xx = Juan's age → xx represents Juan's age
2

Express Maria's age

Maria is 4 years older: x+4x + 4 → Maria's age = x+4x + 4
3

Set up the equation

Sum of ages = 22: x+(x+4)=22x + (x + 4) = 22 → 2x+4=222x + 4 = 22
4

Solve for x

2x=182x = 18, so x=9x = 9 → Juan is 9 years old
5

Find Maria's age

9+4=139 + 4 = 13 → Maria is 13 years old
6

Check the answer

9+13=229 + 13 = 22 ✓ and 13−9=413 - 9 = 4 ✓ → Both conditions satisfied!

Common Mistakes

Forgetting to adjust ALL ages when moving through time

Why it's wrong: When the problem says '5 years ago' or 'in 10 years', you must add or subtract from EVERY person's age, not just one.

Correct: If Juan is xx now and Maria is x+4x + 4 now, then 5 years ago Juan was x−5x - 5 and Maria was (x+4)−5=x−1(x + 4) - 5 = x - 1.

Confusing 'older than' with 'times as old as'

Why it's wrong: '5 years older' means +5, while 'twice as old' means ×2\times 2. These are very different operations!

Correct: Read carefully: 'Maria is 5 years older' → x+5x + 5. 'Maria is twice as old' → 2x2x.

Not checking if the answer makes sense

Why it's wrong: Ages must be positive numbers, and the relationships must hold at all time points mentioned in the problem.

Correct: Always verify: Are all ages positive? Do the original conditions work? If someone was '5 years ago', is their current age at least 5?

Interactive Visual

Ratio Tape Diagram

1:3
Son
12
= 12
Father
12
12
12
= 36
Sum of ages
= 4 parts (48)
Son:1
Father:3
If total is:
Ratio:1:3
Fraction form:1/4 and 3/4
Son value:12
Father value:36

Adjust the ratio parts using + and - buttons.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Lisa is 3 years older than her brother. If her brother is 8 years old, how old is Lisa?

Why It Matters

Age problems appear everywhere in real life:
  • Family planning: Calculating age differences between siblings
  • Eligibility: Determining when someone will be old enough for a license, voting, or retirement
  • Historical research: Figuring out when events happened based on people's ages
  • Legal matters: Calculating ages for inheritance, custody, or insurance purposes
Mastering age problems builds critical thinking skills that apply to many algebraic word problems!

Real World Applications

Family Relationships

Understanding age relationships helps in genealogy research and family planning.

Example:

If a mother is 28 years older than her daughter, and the mother is currently 45, the daughter is 45−28=1745 - 28 = 17 years old.

1Try It Yourself

Two brothers have ages that sum to 30. The older brother is 6 years older than the younger.

How old is each brother?

Step 1: Write the mathematical expression

If younger brother's age = xx, set up the equation:

Retirement Planning

Calculating when someone will reach a certain age is essential for financial planning.

Example:

If you are 42 now and can retire at 65, you have 65−42=2365 - 42 = 23 years until retirement.

2Try It Yourself

A parent is currently 40 and their child is 10. In how many years will the parent be exactly twice the child's age?

Find the number of years until this happens.

Step 1: Write the mathematical expression

Let nn = number of years. Set up the equation:

Key Takeaways

  • 1Define a variable for the unknown age (usually the younger person's age)
  • 2Express other ages in terms of that variable using relationships like 'older' or 'times as old'
  • 3When dealing with past or future ages, adjust ALL ages by adding or subtracting the same number of years
  • 4Set up an equation based on the given condition (sum, difference, ratio, etc.)
  • 5Always check your answer by substituting back into the original problem

Frequently Asked Questions

Usually, choose the younger person's age as xx. This makes other expressions positive (e.g., 'older by 5' becomes x+5x + 5 instead of x−5x - 5).
Usually, choose the younger person's age as xx. This makes other expressions positive (e.g., 'older by 5' becomes x+5x + 5 instead of x−5x - 5).
A negative age means you made an error in setting up the problem, or the problem's conditions are impossible. Re-read the problem and check your equation.
Define one variable for the youngest person, then express all other ages in terms of that variable. You may need multiple equations to solve.

Glossary

Variable
A letter (like xx) used to represent an unknown value
Algebraic expression
A mathematical phrase containing numbers, variables, and operations (e.g., 2x+52x + 5)
Equation
A mathematical statement showing two expressions are equal (e.g., x+5=12x + 5 = 12)
Age difference
The constant number of years between two people's ages, which stays the same over time
Today's challengeOne problem, three tries

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