1. Age Ratios in the Past or Future

What is the question?

These questions usually ask:

“Two people’s ages are in a certain ratio now. What will their ages or ratio be after a few years?”

Or:

“Their ages will be in a certain ratio after 5 years. Find their current ages.”

Idea

When ages are given as a ratio, use x:

Person A = 7x
Person B = 2x

If 5 years pass, add 5 to both:

A = 7x + 5
B = 2x + 5

The important thing is:

Time changes both ages by the same amount.

Example

Question: The present age ratio of A and B is 7:2. After 5 years, their age ratio will be 9:4. Find their present ages.

Step 1: Write their present ages

A = 7x
B = 2x

Step 2: Add 5 years

A = 7x + 5
B = 2x + 5

Step 3: Use the new ratio

(7x + 5) / (2x + 5) = 9 / 4

Cross multiply:

4(7x + 5) = 9(2x + 5)

28x + 20 = 18x + 45

10x = 25

x = 2.5

Step 4: Find their current ages

A = 7 × 2.5 = 17.5 years
B = 2 × 2.5 = 5 years

Answer:

A = 17.5 years
B = 5 years

Shortcut:

Ratio → use x → adjust for years → use the new ratio → solve.

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2. Constant Age Difference

What is the question?

These questions usually ask:

“Two people have an age difference of ___ years. What are their ages?”

Or:

“Their ages were in a certain ratio 10 years ago. Find their present ages.”

Idea

The age difference never changes.

NOW:

Father = 40
Son    = 15

Difference = 25


10 YEARS LATER:

Father = 50
Son    = 25

Difference = 25

Both people get older, but the gap stays the same.

Example

Question: 10 years ago, the ages of A and B were in the ratio 2:3. Their total age today is 70. Find their present ages.

Step 1: Write their ages 10 years ago

A = 2x
B = 3x

Step 2: Move to today

A = 2x + 10
B = 3x + 10

Step 3: Use the total age

(2x + 10) + (3x + 10) = 70

5x + 20 = 70

5x = 50

x = 10

Step 4: Find their ages

A = 2(10) + 10 = 30
B = 3(10) + 10 = 40

Answer:

A = 30 years
B = 40 years

Remember:

Past    → subtract
Future  → add

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3. “When I Was Your Age” / Reciprocal Age Problems

What is the question?

These questions usually sound like:

“When I was as old as you are now, how old were you?”

Or:

“When I was your present age, you were ___ years old.”

They are confusing because they compare different points in time.

Idea

First find the age difference.

Then move both people backward or forward by that amount.

Example

Question: A is 30 years old and B is 20 years old. When A was as old as B is now, how old was B?

Step 1: Find the age difference

A = 30
B = 20

Difference = 30 - 20
           = 10 years

Step 2: Go back 10 years

A: 30 → 20
B: 20 → 10

So:

When A was 20,
B was 10.

Answer: 10 years old.

Visual

10 years ago              NOW

A = 20                    A = 30
B = 10                    B = 20

Remember:

Find the age difference first → move both people by that many years.

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4. Average Age of a Group

What is the question?

These questions usually ask:

“A person joins or leaves a group and the average age changes. Find that person’s age.”

Idea

Use:

Total age = Average × Number of people

Find the total before and after the change.

Example

Question: The average age of 5 people is 30 years. One person leaves, and the average age of the remaining 4 people becomes 28 years. Find the age of the person who left.

Step 1: Find the old total

5 × 30 = 150

Step 2: Find the new total

4 × 28 = 112

Step 3: Find the person’s age

150 - 112 = 38

Answer: 38 years old.

Visual

BEFORE                 AFTER

5 people               4 people
Total = 150            Total = 112

150 - 112 = 38
         ↓
   Person who left

Shortcut:

Old total → New total → Difference.

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5. Fraction / Multiple of Another Person’s Age

What is the question?

These questions say things like:

“A is 3/2 times B’s age 4 years ago.”

Or:

“A is twice B’s age 5 years from now.”

Idea

First adjust the age for time. Then apply the fraction or multiple.

Years ago   → subtract
Years later → add

Example

Question: A’s age is 3/2 of B’s age 4 years ago. B is currently 20. Find A’s age.

Step 1: Adjust B’s age

20 - 4 = 16

Step 2: Apply the fraction

A = 3/2 × 16
A = 24

Answer: A is 24 years old.

Remember:

Adjust first → apply the fraction/multiple second.

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6. Father-Son-Grandchild / Generational Problems

What is the question?

These questions give age differences between generations and ask:

“Find the ages of the father, son and grandchild.”

Idea

Start with the youngest person, then build upward.

Example

Question: A father is 30 years older than his son. The son is 20 years older than his daughter. Their total age is 120 years. Find all three ages.

Step 1: Start with the daughter

Daughter = x

Step 2: Find the son

Son = x + 20

Step 3: Find the father

Father = x + 20 + 30
       = x + 50

Step 4: Use the total

x + (x + 20) + (x + 50) = 120

3x + 70 = 120

3x = 50

x = 16⅔

Again, the numbers give fractional ages, so this is mainly demonstrating the method. In an actual exam, the values are generally chosen to give sensible ages.

Visual

Daughter
   ↓ +20
  Son
   ↓ +30
Father

Remember:

Start with the youngest → add the differences → use the total.

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7. Present Age Ratio

What is the question?

These are the simplest ratio questions:

“The present ages of A and B are in the ratio 5:3. Their total age is 64. Find their ages.”

Idea

A ratio tells you the number of parts.

A : B = 5 : 3

A = 5 parts
B = 3 parts

Example

Question: A and B’s ages are in the ratio 5:3. Their combined age is 64. Find their ages.

Step 1: Add the parts

5 + 3 = 8

Step 2: Find one part

64 ÷ 8 = 8

Step 3: Find both ages

A = 5 × 8 = 40
B = 3 × 8 = 24

Answer:

A = 40 years
B = 24 years

Shortcut:

Add ratio parts → divide total → multiply each part.

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8. Age Sum and Age Difference

What is the question?

These questions give the sum and difference of two people’s ages.

For example:

“The sum of A and B’s ages is 50, and their age difference is 10. Find their ages.”

Idea

If:

Older + Younger = Sum
Older - Younger = Difference

Then:

Older = (Sum + Difference) / 2

Younger = (Sum - Difference) / 2

Example

Question: The sum of A and B’s ages is 50 years. A is 10 years older than B. Find their ages.

Step 1: Find A

A = (50 + 10) / 2
  = 60 / 2
  = 30

Step 2: Find B

B = (50 - 10) / 2
  = 40 / 2
  = 20

Answer:

A = 30
B = 20

Visual

A + B = 50
A - B = 10

        ↓

A = 30
B = 20

Shortcut:

Older = (Sum + Difference) ÷ 2 Younger = (Sum - Difference) ÷ 2

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9. Multiple People with Age Ratios

What is the question?

These questions involve 3 or more people whose ages are given in ratios.

For example:

“The ages of A, B and C are in the ratio 2:3:5. Their total age is 50. Find their ages.”

Idea

Treat every ratio number as a part.

Example

Question: A, B and C’s ages are in the ratio 2:3:5. Their total age is 50. Find their ages.

Step 1: Add all parts

2 + 3 + 5 = 10 parts

Step 2: Find one part

50 ÷ 10 = 5

Step 3: Find each age

A = 2 × 5 = 10
B = 3 × 5 = 15
C = 5 × 5 = 25

Answer:

A = 10
B = 15
C = 25

Visual

A: [ ][ ]          = 2 parts
B: [ ][ ][ ]       = 3 parts
C: [ ][ ][ ][ ][ ] = 5 parts

Total = 10 parts

Remember:

Add ALL ratio parts → find one part → multiply.

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10. Age Ratio at Two Different Points in Time

What is the question?

These questions give two ratios at two different times.

For example:

“The ages of A and B are in the ratio 3:2 now. After 5 years, their ages will be in the ratio 4:3. Find their present ages.”

Idea

Write their current ages using x, then move both ages forward.

Example

Question: A and B’s present ages are in the ratio 3:2. After 5 years, the ratio will be 4:3. Find their present ages.

Step 1: Write present ages

A = 3x
B = 2x

Step 2: Add 5 years

A = 3x + 5
B = 2x + 5

Step 3: Use the future ratio

(3x + 5) / (2x + 5) = 4 / 3

Cross multiply:

3(3x + 5) = 4(2x + 5)

9x + 15 = 8x + 20

x = 5

Step 4: Find their ages

A = 3 × 5 = 15
B = 2 × 5 = 10

Answer:

A = 15 years
B = 10 years

Shortcut:

Current ratio → use x → move through time → apply second ratio.

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11. Person Joins or Leaves a Group

What is the question?

These questions are slightly different from basic average problems.

They ask:

“A new person joins a group and the average age increases. Find the age of the new person.”

Or:

“A person leaves and the average decreases. Find that person’s age.”

Idea

Again, use:

Total = Average × Number of people

Then compare the old and new totals.

Example

Question: The average age of 6 people is 25 years. A new person joins the group, and the average becomes 27 years. Find the new person’s age.

Step 1: Find the old total

6 × 25 = 150

Step 2: Find the new total

There are now 7 people:

7 × 27 = 189

Step 3: Find the new person’s age

189 - 150 = 39

Answer: 39 years old.

Visual

BEFORE                 AFTER

6 people               7 people
Total = 150            Total = 189

189 - 150 = 39
         ↓
    New person's age

Remember:

New total - Old total = Age of person who joined.

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12. Mixed Age Problems

What is the question?

These are the more difficult competitive-exam questions.

They combine two or more ideas, such as:

• ratio + time
• difference + ratio
• average + age difference
• multiple people + ratios
• past ratio + future ratio

Idea

Don’t look for a special formula.

Break the question into smaller pieces.

Example

Question: The present ages of A and B are in the ratio 3:2. Five years ago, their total age was 35. Find their present ages.

Step 1: Use the present ratio

A = 3x
B = 2x

Step 2: Go 5 years into the past

A = 3x - 5
B = 2x - 5

Step 3: Use the past total

(3x - 5) + (2x - 5) = 35

Solve:

5x - 10 = 35

5x = 45

x = 9

Step 4: Find present ages

A = 3 × 9 = 27
B = 2 × 9 = 18

Answer:

A = 27 years
B = 18 years

How to recognize a mixed problem

If a question contains several clues:

ratio + years + total

or:

ratio + difference + past/future

it is probably a mixed age problem.

Remember:

Don’t search for one magic formula. Identify each piece and solve it step by step.

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13. Age Difference Between Three or More People

What is the question?

These questions give differences between several people.

For example:

“A is 5 years older than B, and B is 8 years older than C. If C is 20, find A’s age.”

Idea

Build the ages one person at a time.

Example

Question: A is 5 years older than B. B is 8 years older than C. C is 20 years old. Find A and B’s ages.

Step 1: Find B

B = 20 + 8
  = 28

Step 2: Find A

A = 28 + 5
  = 33

Answer:

C = 20
B = 28
A = 33

Visual

C = 20
   ↓ +8
B = 28
   ↓ +5
A = 33

Remember:

Start from the person whose age is known → move up or down using the differences.

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14. Age Problems with a Fixed Future/Past Relationship

What is the question?

These questions ask something like:

“After 10 years, A will be twice B’s age. Their present age difference is 20. Find their current ages.”

Idea

Use the age difference and the future relationship together.

Example

Question: After 10 years, A will be twice B’s age. A is currently 20 years older than B. Find their present ages.

Step 1: Use the age difference

If B is x:

B = x
A = x + 20

Step 2: Move 10 years forward

A = x + 30
B = x + 10

Step 3: Use the future relationship

A will be twice B:

x + 30 = 2(x + 10)

Solve:

x + 30 = 2x + 20

x = 10

Step 4: Find present ages

B = 10
A = 30

Answer:

A = 30 years
B = 10 years

Remember:

Use the age difference to create the ages → adjust time → apply the given relationship.

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