Problems on Ages questions with answers and concepts

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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 and rules

6 rules

Time shift property: If present age is $x$ , age $n$ years ago was $x - n$ , and age $n$ years hence will be $x + n$

Age ratio representation: If $A : B = m : n$ , then $A = mk$ and $B = nk$ for some multiplier $k > 0$

Sum partition: If $A + B = S$ and $A : B = m : n$ , then $A = S \times \frac{m}{m + n}$ and $B = S \times \frac{n}{m + n}$

Invariant difference: The difference in ages $(A - B)$ remains constant across past, present, and future

Multiple of age relation: If $A$ is $n$ times $B$ after $t$ years, $A + t = n (B + t)$

Average age: $Average Age = \frac{Sum of ages}{Number of people}$

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.

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Worked examples

Example 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

Step 1: Express present ages with multiplier $x$ .
- Age of A = $5 x$
- Age of B = $7 x$

Step 2: Set up ratio equation for eight years ago.

\frac{5 x - 8}{7 x - 8} = \frac{2}{3}

3 (5 x - 8) = 2 (7 x - 8)

15 x - 24 = 14 x - 16 ⟹ 15 x - 14 x = 24 - 16 ⟹ x = 8

Step 3: Calculate present ages.

Age of A = 5 \times 8 = 40 years

Age of B = 7 \times 8 = 56 years

Answer: A is 40 years old and B is 56 years old.

Example 2

A father is currently three times as old as his son. In 5 years, he will be 2.5 times as old as his son. What are their present ages?

Solution

Step 1: Let the son's present age be $S$ . Father's age $F = 3 S$ .

Step 2: Formulate equation 5 years hence.

F + 5 = 2.5 (S + 5)

3 S + 5 = 2.5 S + 12.5

0.5 S = 7.5 ⟹ S = \frac{7.5}{0.5} = 15 years

Step 3: Father's present age:

F = 3 \times 15 = 45 years

Answer: Father is 45 years old and son is 15 years old.

Example 3

The sum of the present ages of a father and son is 60 years. Six years ago, the father was five times as old as the son. What will be the son's age 6 years from now?

Solution

Step 1: Express ages 6 years ago.

(F - 6) + (S - 6) = 60 - 12 = 48 years

Step 2: Use the past ratio $(F - 6) = 5 (S - 6)$ .

5 (S - 6) + (S - 6) = 48 ⟹ 6 (S - 6) = 48 ⟹ S - 6 = 8

S = 14 years (present age)

Step 3: Age 6 years in the future.

S_{future} = 14 + 6 = 20 years

Answer: 20 years.

Example 4

The sum of the ages of 5 siblings born at regular intervals of 3 years is 50 years. What is the age of the youngest child?

Solution

Step 1: Express sibling ages in arithmetic progression with difference $d = 3$ .

x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50

Step 2: Solve for youngest age $x$ .

5 x + 30 = 50 ⟹ 5 x = 20 ⟹ x = 4 years

Answer: 4 years old.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The present ages of A and B are in the ratio 4:5. Five years ago, their ages were in the ratio 3:4. What will be the ratio of their ages after 10 years?

Answer: B. 6:7

Explanation

Step 1: Let present ages be 4x and 5x.
Step 2: Five years ago: $(4 x - 5) / (5 x - 5) = 3/4$
Step 3: 4(4x - 5) = 3(5x - 5)
Step 4: $16 x - 20 = 15 x - 15$
Step 5: $x = 5$
Step 6: Present ages: 20 and 25 years.
Step 7: After 10 years: 30 and 35 years.
Step 8: Ratio = 30: $35 = 6$ :7
Answer: 6:7

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

The ratio between the present ages of A and B is 5:3 respectively. The ratio between A's age 4 years ago and B's age 4 years hence is 1:1. What is the ratio between A's age 4 years hence and B's age 4 years ago?

Answer: B. 3:1

Explanation

Step 1: Let present ages be 5x and 3x.
Step 2: A's age 4 years $a g o = 5 x - 4$
Step 3: B's age 4 years hence = 3x + 4
Step 4: $(5 x - 4) / (3 x + 4) = 1/1$
Step 5: $5 x - 4 = 3 x + 4$
Step 6: $2 x = 8$ , $x = 4$
Step 7: A's present age = 20, B's present age = 12
Step 8: A's age 4 years hence = 24, B's age 4 years $a g o = 8$
Step 9: Ratio = 24: $8 = 3$ :1
Answer: 3:1

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The ratio of the ages of a mother and her daughter is 9:4. If the mother was 25 years old when the daughter was born, what is the present age of the daughter?

15 years
18 years
20 years
24 years

Answer: C. 20 years

Explanation

Step 1: Let ages be 9x and 4x.
Step 2: Difference in age = 9x - 4x = 5x
Step 3: Age difference when daughter was born = Mother's age = 25 years.
Step 4: $5 x = 25$
Step 5: $x = 5$
Step 6: Daughter's present age = 4 x $5 = 20$ years.
Answer: 20 years

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

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.

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Key concepts

Present, Past, and Future Ages

Ratio-Based Age Problems

Constant Age Difference

Multiple Time Period Relationships

Group Age Problems

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 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

Step 1: Express present ages with multiplier $x$ .
- Age of A = $5 x$
- Age of B = $7 x$

Step 2: Set up ratio equation for eight years ago.

\frac{5 x - 8}{7 x - 8} = \frac{2}{3}

3 (5 x - 8) = 2 (7 x - 8)

15 x - 24 = 14 x - 16 ⟹ 15 x - 14 x = 24 - 16 ⟹ x = 8

Step 3: Calculate present ages.

Age of A = 5 \times 8 = 40 years

Age of B = 7 \times 8 = 56 years

Answer: A is 40 years old and B is 56 years old.

Example 2

A father is currently three times as old as his son. In 5 years, he will be 2.5 times as old as his son. What are their present ages?

Solution

Step 1: Let the son's present age be $S$ . Father's age $F = 3 S$ .

Step 2: Formulate equation 5 years hence.

F + 5 = 2.5 (S + 5)

3 S + 5 = 2.5 S + 12.5

0.5 S = 7.5 ⟹ S = \frac{7.5}{0.5} = 15 years

Step 3: Father's present age:

F = 3 \times 15 = 45 years

Answer: Father is 45 years old and son is 15 years old.

Example 3

The sum of the present ages of a father and son is 60 years. Six years ago, the father was five times as old as the son. What will be the son's age 6 years from now?

Solution

Step 1: Express ages 6 years ago.

(F - 6) + (S - 6) = 60 - 12 = 48 years

Step 2: Use the past ratio $(F - 6) = 5 (S - 6)$ .

5 (S - 6) + (S - 6) = 48 ⟹ 6 (S - 6) = 48 ⟹ S - 6 = 8

S = 14 years (present age)

Step 3: Age 6 years in the future.

S_{future} = 14 + 6 = 20 years

Answer: 20 years.

Example 4

The sum of the ages of 5 siblings born at regular intervals of 3 years is 50 years. What is the age of the youngest child?

Solution

Step 1: Express sibling ages in arithmetic progression with difference $d = 3$ .

x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50

Step 2: Solve for youngest age $x$ .

5 x + 30 = 50 ⟹ 5 x = 20 ⟹ x = 4 years

Answer: 4 years old.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The present ages of A and B are in the ratio 4:5. Five years ago, their ages were in the ratio 3:4. What will be the ratio of their ages after 10 years?

Answer: B. 6:7

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

Answer: B. 3:1

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The ratio of the ages of a mother and her daughter is 9:4. If the mother was 25 years old when the daughter was born, what is the present age of the daughter?

15 years
18 years
20 years
24 years

Answer: C. 20 years

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

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.

Key formulas and rules

Key concepts

Present, Past, and Future Ages

Ratio-Based Age Problems

Constant Age Difference

Multiple Time Period Relationships

Group Age Problems

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Present, Past, and Future Ages

Ratio-Based Age Problems

Constant Age Difference

Multiple Time Period Relationships

Group Age Problems

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.