About Problems on Ages
Age questions look like riddles, but each one is a linear equation in a single unknown. There is only one special rule, and it decides most of the questions: everybody grows older at the same rate, so the gap between two ages never changes, however many years pass.
What you need to understand
- The difference between two ages is constant. If A is 10 years older than B today, A will still be 10 years older in twenty years.
- Ages are compared at one moment at a time: now, n years ago, or n years later. Mixing two moments in one equation is the commonest error.
- A statement such as A will be twice as old as B is an equation about the two ages at that future moment, not about their present ages.
- Ratios of ages change with time even though differences do not, because both ages grow by the same amount rather than by the same proportion.
- You may take either age as the unknown. Choosing whichever makes the equation simplest is a free choice.
Formulas to remember
How to work through these questions
- Write both ages at the moment the comparison is made, using a single unknown.
- Read the wording closely. After n years adds n to both ages, and n years ago subtracts n from both.
- Turn the multiple or the ratio into an equation and solve it for the unknown.
- Answer the question actually asked. If the unknown was the elder and the question wants the younger, convert.
- Substitute the answer back, because an age question almost always contains a condition you can test.
Mistakes that cost marks
- Adding the years to one person and not to the other.
- Giving the age difference itself as the answer instead of an age.
- Applying a future ratio to the present ages.
- Setting up a fresh age difference for the future instead of using the one that never changes.
- Allowing the unknown to go negative when the years are subtracted from an age.
Worked example
A father is three times as old as his son. In 12 years he will be twice as old. Find the present age of the son.
- Let the son be x years old, so the father is 3x years old.
- In 12 years the son will be x + 12 and the father will be 3x + 12.
- The father will then be twice the son: 3x + 12 = 2(x + 12) = 2x + 24.
- Solving gives x = 12, so the son is 12 years old.
- Check: the father is 36 now, and in 12 years the two ages are 24 and 48, which is exactly double.
Answer: 12 years
Practice questions with answers
A few Problems on Ages questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
The total of the ages of A and B is 72 years, and the difference of their ages is 16 years. Find the elder one's age.
Answer: Option A — with explanation
If B is b years old then A is b + 16, and together they are 72.
So 2b + 16 = 72, giving b = 28 years.
A's age = 28 + 16 = 44 years.
Common mistakes
- gave the younger person's age: 28
- split the total equally and ignored the age difference: 36
- added the whole difference on top of half the total: 52
Question 2
The present ages of A and B are in the ratio 7 : 8. After 5 years the ratio of their ages becomes 47 : 53. Find A's present age.
Answer: Option C — with explanation
Let the present ages be 7x and 8x years.
After 5 years the ratio is 47 : 53, so 53(7x + 5) = 47(8x + 5).
Solving gives x = 6, so A's present age is 7 x 6 = 42 years.
Common mistakes
- gave A's age after the years have passed: 47
- added the two present ages together: 90
- subtracted the years instead of adding them: 37
Question 3
A is older than B by 33 years. 8 years from now, A's age will be double B's age. What is B's age today?
Answer: Option B — with explanation
Let B's present age be b years, so A is b + 33 today.
After 8 years A is b + 41 and B is b + 8.
A is twice B: b + 41 = 2(b + 8) = 2b + 16.
So b = 33 - 8 = 25 years.
Common mistakes
- added the age difference and the number of years instead of subtracting: 41
- subtracted the number of years twice over: 17
- gave the age difference itself instead of an age: 33
Question 4
A and B together are 46 years old. A is 16 years older than B. How old is A?
Answer: Option C — with explanation
If B is b years old then A is b + 16, and together they are 46.
So 2b + 16 = 46, giving b = 15 years.
A's age = 15 + 16 = 31 years.
Common mistakes
- gave the younger person's age: 15
- gave the difference of their ages instead of an age: 16
- split the total equally and ignored the age difference: 23
Question 5
A and B are now in the ratio 1 : 4. 7 years later the ratio of their ages will be 31 : 103. What is A's age today?
Answer: Option B — with explanation
Let the present ages be 2x and 8x years.
After 7 years the ratio is 31 : 103, so 103(2x + 7) = 31(8x + 7).
Solving gives x = 12, so A's present age is 2 x 12 = 24 years.
Common mistakes
- gave A's age after the years have passed: 31
- subtracted the years instead of adding them: 17
- added the two present ages together: 120
Frequently asked questions
Why does the age difference never change?
Because both people become one year older each year. The gap between them therefore stays exactly the same, even though the ratio of their ages keeps changing.
What does n years hence mean?
It means n years from now, so n is added to both ages. The phrase n years ago means n is subtracted from both ages.
Can an age come out as a fraction?
It should not. Ages in these questions are whole numbers, so a fractional age is a signal that the equation was set up wrongly.
How do I decide which age to take as the unknown?
Use the age the question asks for when you can. If the question gives a ratio, letting the unknown be the multiplier of the ratio usually keeps the arithmetic simplest.
