Age Problems
Learn to solve word problems involving ages, including present ages, past ages, and future ages.
Definition
- " years ago" means subtract from current age
- "In years" means add to current age
- "Twice as old" means multiply by
- " years older" means add to the other person's age
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Worked Examples
Maria is 4 years older than her brother Juan. The sum of their ages is 22. How old is each person?
Define the variable
Let = Juan's age → represents Juan's age
Express Maria's age
Maria is 4 years older: → Maria's age =
Set up the equation
Sum of ages = 22: →
Solve for x
, so → Juan is 9 years old
Find Maria's age
→ Maria is 13 years old
Check the answer
✓ and ✓ → Both conditions satisfied!
Answer: Juan is 9 years old and Maria is 13 years old.
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 now and Maria is now, then 5 years ago Juan was and Maria was .
Confusing 'older than' with 'times as old as'
Why it's wrong: '5 years older' means +5, while 'twice as old' means . These are very different operations!
Correct: Read carefully: 'Maria is 5 years older' → . 'Maria is twice as old' → .
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?
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Practice Problems
15 problemsLisa is 3 years older than her brother. If her brother is 8 years old, how old is Lisa?
Why It Matters
- 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
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 years old.
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 = , 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 years until retirement.
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 = 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
Glossary
- Variable
- A letter (like ) used to represent an unknown value
- Algebraic expression
- A mathematical phrase containing numbers, variables, and operations (e.g., )
- Equation
- A mathematical statement showing two expressions are equal (e.g., )
- Age difference
- The constant number of years between two people's ages, which stays the same over time