Age Problems
Translate past, present, and future age relationships into equations and solve multi-condition UPCAT-style age problems.
Age Problems
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Put every age on one timeline
Choose one variable for a person’s present age. Express every other age from it.
t years ago: present age − tt years from now: present age + tIf one person is 6 years older, that difference stays 6 in the past and future. A ratio such as 2:1 usually changes because equal years are added to both ages.
Present age
Define variables at the same point in time.
Years ago
Subtract the same number from every person involved.
Years from now
Add the same number to every person involved.
Age difference
The difference between two ages stays constant.
Age ratio
The ratio generally changes over time.
Make a three-column timeline
Write three headings: Past | Present | Future. Place each person’s age under the correct heading before forming an equation.
For “in t years, A will be twice B,” write:
A + t = 2(B + t)Do not write A + t = 2B + t; the time change belongs inside the multiplication.
Why it works
Most age-problem errors come from comparing ages at different times or addingsubtracting years from only one person.
Five forms you should recognize
Problem: Mara is 6 years older than Liza. Their present ages total 30. Find Mara’s age.
Setup: Let Liza = x, so Mara = x + 6.
x + (x + 6) = 302x = 24 → x = 12Mara is 12 + 6 = 18.
Problem: A father is 36 and his son is 12. In how many years will the father be twice as old?
Setup: Both ages increase by t.
36 + t = 2(12 + t)36 + t = 24 + 2t → t = 12Check: In 12 years they will be 48 and 24.
Problem: Ana is 24 and Bea is 16. How many years ago was Ana twice Bea’s age?
24−t = 2(16−t)24−t = 32−2t → t = 8Check: Eight years ago they were 16 and 8.
Problem: Two siblings’ ages are in the ratio 3:5 and differ by 8 years. Find the older age.
The difference of 2 ratio-parts equals 8, so one part is 4.
older age = 5(4) = 20Problem: Five years ago, Carlo was half Dana’s age. In 5 years, their ages will total 50.
Let their present ages be C and D.
C−5 = (D−5)2(C + 5) + (D + 5) = 50The second equation gives C + D = 40. Solving the system gives C = 15 and D = 25.
Check before you commit
- Adding years to only one person
- Multiplying before placing both future ages in parentheses
- Assuming an age ratio stays constant
- Changing the age difference over time
- Using “older than twice” as 2(x + k) instead of 2x + k
- Comparing one person’s past age with another’s present age
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Age Problems FAQ
Does an age difference ever change?
No. If one person is 8 years older now, the difference was and will remain 8 years.
Does an age ratio stay the same?
Usually not. Equal additions change a ratio unless the ages are equal.
How do I check my answer?
Substitute the present ages into every past or future condition in the problem.
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