Problems on Ages

Present, past and future age puzzles

Practice →

Problems on ages is a classic quantitative aptitude topic that involves calculating ages of individuals based on relationships given at different points in time (present, past, or future). These problems require setting up equations based on age ratios and solving for unknown ages using algebraic methods.

Key Formulas

If the present age of a person is x years, then: n years ago their age was (x - n) years, and n years hence their age will be (x + n) years.\text{If the present age of a person is x years, then: n years ago their age was (x - n) years, and n years hence their age will be (x + n) years.}
Age ratio A : B = m : n means A’s age / B’s age = m / n, or A’s age = m x k and B’s age = n x k for some positive constant k.\text{Age ratio A : B = m : n means A's age / B's age = m / n, or A's age = m x k and B's age = n x k for some positive constant k.}
If the sum of ages of two people is S and their ratio is m : n, then their ages are (m x S)/(m + n) and (n x S)/(m + n).\text{If the sum of ages of two people is S and their ratio is m : n, then their ages are (m x S)/(m + n) and (n x S)/(m + n).}
The difference between ages remains constant over time: if A is 5 years older than B today, A will always be 5 years older than B.\text{The difference between ages remains constant over time: if A is 5 years older than B today, A will always be 5 years older than B.}
If A’s age is n times B’s age today, after/fefore x years: (A +/- x) = n x (B +/- x) - adjust signs based on past/future.\text{If A's age is n times B's age today, after/fefore x years: (A +/- x) = n x (B +/- x) - adjust signs based on past/future.}
Average age of a group = (Sum of all ages) / (Number of people in the group).\text{Average age of a group = (Sum of all ages) / (Number of people in the group).}

Key Concepts

Present, Past, and Future Ages

Always define variables for present ages. If a problem mentions '5 years ago', subtract 5 from the present age. If it mentions 'after 8 years' or '8 years hence', add 8 to the present age. Be careful with double negatives: '5 years before 3 years ago' means 8 years ago from present.

Ratio-Based Age Problems

When ages are given in ratio m:n, represent the ages as mx and nx where x is a common multiplier. Set up equations using other given information (sum of ages, difference, or relationship after certain years) to solve for x, then calculate actual ages.

Constant Age Difference

The age gap between two people never changes. If Person A is 10 years older than Person B, this remains true 5 years ago, today, or 20 years in the future. Use this property to simplify problems involving multiple time periods.

Multiple Time Period Relationships

Some problems state that '5 years ago, A was twice as old as B'. This translates to: (A's present age - 5) = 2 x (B's present age - 5). Always express such relationships using present age variables for consistency.

Group Age Problems

For problems involving families or groups, track each person's age separately. When a new member is born, their age starts at 0. The average age of a group changes when members are added or when time passes -- account for all members when calculating averages.

Solved Examples

Problem 1:

The present ages of A and B are in the ratio 5:7. Eight years ago, the ratio of their ages was 2:3. Find their present ages.

Solution:

Let the present ages of A and B be 5x and 7x years respectively.
Eight years ago, A's age was (5x - 8) and B's age was (7x - 8).
Given that eight years ago, the ratio was 2:3.
So: (5x - 8) / (7x - 8) = 2 / 3
Cross-multiply: 3(5x - 8) = 2(7x - 8)
15x - 24 = 14x - 16
15x - 14x = 24 - 16
x = 8
Therefore, A's present age = 5 x 8 = 40 years
B's present age = 7 x 8 = 56 years
Verification: Eight years ago, A was 32 and B was 48. Ratio 32:48 = 2:3

Problem 2:

A father is 4 times as old as his son. In 20 years, he will be only twice as old as his son. Find their present ages.

Solution:

Let the son's present age be x years.
Then the father's present age is 4x years.
In 20 years: Son's age = x + 20, Father's age = 4x + 20
Given: In 20 years, father will be twice as old as son.
So: 4x + 20 = 2(x + 20)
4x + 20 = 2x + 40
4x - 2x = 40 - 20
2x = 20
x = 10
Therefore, Son's present age = 10 years
Father's present age = 4 x 10 = 40 years
Verification: In 20 years, son will be 30 and father will be 60. 60 = 2 x 30

Problem 3:

The sum of the ages of a father and his son is 50 years. Five years ago, the father was 7 times as old as his son. Find the present age of the son.

Solution:

Let the son's present age be x years.
Then the father's present age = (50 - x) years.
Five years ago: Son's age = x - 5, Father's age = 50 - x - 5 = 45 - x
Given: Five years ago, father was 7 times as old as son.
So: 45 - x = 7(x - 5)
45 - x = 7x - 35
45 + 35 = 7x + x
80 = 8x
x = 10
Therefore, the son's present age is 10 years.
Father's present age = 50 - 10 = 40 years.
Verification: Five years ago, son was 5 and father was 35. 35 = 7 x 5

Problem 4:

The average age of a husband and wife was 25 years when they married 6 years ago. Now the average age of the husband, wife, and their child is 22 years. Find the present age of the child.

Solution:

Six years ago, average age of husband and wife was 25 years.
Sum of their ages 6 years ago = 25 x 2 = 50 years.
Present sum of husband and wife ages = 50 + 6 + 6 = 62 years.
Let child's present age be x years.
Now, average of all three = 22 years.
So: (62 + x) / 3 = 22
62 + x = 66
x = 4
Therefore, the child's present age is 4 years.

Tips & Tricks

  • Always define variables for present ages and express all other ages relative to present time.
  • Use the constant age difference property to quickly eliminate options in multiple-choice questions.
  • When dealing with ratios, introduce a common multiplier (like x) to represent actual ages.
  • For complex problems with multiple time periods, consider creating a timeline diagram.
  • Verify your answer by plugging it back into the original conditions of the problem.
  • Remember that age cannot be negative -- if your solution gives a negative age, recheck your calculations.

Ready to practice?

Test your understanding with questions and get instant feedback.

Start Exercise →