Problems on Ages made clear.
Put everyone on the same timeline. Each person ages by the same number of years, so age differences stay constant even though ratios change.
Key concepts in Problems on Ages
Future and past
If present age is x, age after t years is x+t; age t years ago is x−t.
Constant difference
For two people, their age difference does not change over time.
Ratio model
Present ages in ratio a:b can be written ak and bk. Add years to both actual ages, not to the ratio terms alone.
Validity
A past-age statement requires nonnegative ages at that time. Reject roots that contradict the story.
Age timeline: mother and child
| Person | Present | After t years | After 12 years |
|---|---|---|---|
| Mother | 36 | 36+t | 48 |
| Child | 12 | 12+t | 24 |
| Difference | 24 | 24 | 24 |
A reliable solving method
- Define variables for present ages.
- Create a small past–present–future table.
- Translate each ratio or sum into an equation.
- Substitute the final ages into every condition.
Step-by-step solved examples
A is 4 years older than B. Their total age is 30. Find both.
- Let B = x and A = x+4.
- 2x+4 = 30, so x = 13.
- B is 13 and A is 17.
Present ages are 2:3. After 5 years they are 3:4.
- Write ages as 2k and 3k.
- (2k+5)/(3k+5)=3/4 gives 8k+20=9k+15.
- k=5, so ages are 10 and 15; after 5 years they are 15:20.
A mother is 36 and her child is 12. When will she be twice the child’s age?
- After t years: mother 36+t, child 12+t.
- 36+t = 2(12+t).
- t=12; their ages will be 48 and 24.
Problems on Ages: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
A is 18 now. Age 7 years ago?
Show answer and explanation — question 1
Answer C: 11
18−7 = 11.
Two ages total 40 and differ by 8. Younger age?
Show answer and explanation — question 2
Answer B: 16
(40−8)/2 = 16.
Ages are 3:5 and total 48. Older age?
Show answer and explanation — question 3
Answer D: 30
5/8 × 48 = 30.
Age gap is 6 now. Gap after 10 years?
Show answer and explanation — question 4
Answer B: 6
Both gain 10 years, so the difference stays 6.
Father 40, child 10. After how many years is father three times child?
Show answer and explanation — question 5
Answer A: 5
40+t=3(10+t), giving t=5.
A is twice B now. B is 12. Their total after 3 years?
Show answer and explanation — question 6
Answer C: 42
A=24; after 3 years total is 27+15 = 42.
Frequently asked questions
Does an age ratio stay fixed?
Usually not. Adding the same positive amount to different ages changes their ratio.
Can an answer be a fraction of a year?
Yes, unless the question requires completed whole years. Convert carefully if months are requested.
Define variables for present ages. Substitute the final ages into every condition.