Age Problem | CSE Reviewer

What Is an Age Problem?

An age problem tells you about people's ages at different times (now, before, or after) and asks you to find their age now or at another time.

For example: "John is twice as old as Mary. In 5 years, John will be 30. How old is Mary now?"

Tip: Think: Age problems use "now," "before" (past), and "after" (future).

Keywords:
- "Now" = current age
- "Before" / "ago" = past (subtract years)
- "After" / "in" / "will be" = future (add years)

Meaning: Age problems give you information about ages at different times and ask you to find a specific age.

Key Rules for Age Problems

There are 3 important rules to remember:

• Rule 1: When someone gets older, everyone gets older by the same number of years.
• Rule 2: The difference between ages stays the same forever.
• Rule 3: Use variables (like x) for unknown ages.

Rule 1 Example: If John is 5 years older than Mary now, he will still be 5 years older in 10 years.

Rule 2 Example: If Mom is 30 now and baby is 0, the difference is 30. When baby is 10, Mom will be 40. Difference is still 30.

Rule 3 Example: If you don't know John's age, write it as "x" or "J."

Meaning: All people age by the same amount, age differences stay constant, and use variables for unknown ages.

How to Solve Age Problems (Step-by-Step)

Follow these steps to solve any age problem:

• Step 1: Identify what you need to find (who and when)
• Step 2: Let a variable (like x) represent the unknown age "now"
• Step 3: Write expressions for "before" (subtract years) and "after" (add years)
• Step 4: Write an equation using the information given
• Step 5: Solve the equation
• Step 6: Check your answer

Shortcut: Always start with "now" as your variable. Past = variable - years. Future = variable + years.

Type 1: One Person, Past and Future

This type has one person and talks about their age in the past or future.

Example 1: In 7 years, Anna will be 25. How old is she now?

Step 1: What to find? → Anna's age now
Step 2: Let x = Anna's age now
Step 3: In 7 years → x + 7
Step 4: Equation → x + 7 = 25
Step 5: Solve → x = 25 - 7 = 18

Answer: Anna is 18 years old now

Example 2: 5 years ago, Ben was 12. How old is he now?

Step 1: What to find? → Ben's age now
Step 2: Let x = Ben's age now
Step 3: 5 years ago → x - 5
Step 4: Equation → x - 5 = 12
Step 5: Solve → x = 12 + 5 = 17

Answer: Ben is 17 years old now

Example 3: In 10 years, Maria will be twice as old as she is now. How old is she now?

Step 1: What to find? → Maria's age now
Step 2: Let x = Maria's age now
Step 3: In 10 years → x + 10
Step 4: "Twice as old as now" → 2x
Step 5: Equation → x + 10 = 2x
Step 6: Solve → 10 = 2x - x → 10 = x

Answer: Maria is 10 years old now

Meaning: For one person, past = variable - years, future = variable + years. Set up equation and solve.

Type 2: Two People, Same Time

This type has two people and compares their ages at the same time (usually "now").

Example 1: John is twice as old as Mary. Together, their ages add to 30. How old is each?

Step 1: What to find? → John's age and Mary's age now
Step 2: Let x = Mary's age now
Step 3: John is twice → John = 2x
Step 4: Together = 30 → x + 2x = 30
Step 5: Solve → 3x = 30 → x = 10
Step 6: Mary = 10, John = 2 × 10 = 20

Answer: Mary is 10, John is 20

Example 2: Alice is 8 years older than Bob. Alice is 25. How old is Bob?

Step 1: What to find? → Bob's age now
Step 2: Let x = Bob's age now
Step 3: Alice is 8 older → Alice = x + 8
Step 4: Alice = 25 → x + 8 = 25
Step 5: Solve → x = 25 - 8 = 17

Answer: Bob is 17 years old

Meaning: For two people at the same time, express one in terms of the other, then set up equation.

Type 3: Two People, Different Times

This type has two people and compares their ages at different times (one now, one in past/future).

Example 1: Father is 3 times as old as son. In 5 years, Father will be 2 times as old as son. How old is each now?

Step 1: Let x = son's age now
Step 2: Father = 3x
Step 3: In 5 years:
- Son = x + 5
- Father = 3x + 5
Step 4: Father = 2 × son → 3x + 5 = 2(x + 5)
Step 5: Solve:
3x + 5 = 2x + 10
3x - 2x = 10 - 5
x = 5
Step 6: Son = 5, Father = 3 × 5 = 15

Answer: Son is 5, Father is 15

Example 2: Anna is 12. 3 years ago, she was twice as old as Ben. How old is Ben now?

Step 1: Let x = Ben's age now
Step 2: 3 years ago:
- Anna = 12 - 3 = 9
- Ben = x - 3
Step 3: Anna = 2 × Ben → 9 = 2(x - 3)
Step 4: Solve:
9 = 2x - 6
9 + 6 = 2x
15 = 2x
x = 7.5

Answer: Ben is 7.5 years old (or 7 years and 6 months)

Meaning: For two people at different times, write expressions for each time, then set up equation.

Tips and Shortcuts for Age Problems

Age problems can be fast if you use shortcuts. Here are the best tips.

Tips:

• Start with "now" as your variable (x).
• Past = x - years, Future = x + years.
• Age difference stays the same forever.
• "Twice" = 2×, "Three times" = 3×, "Half" = 0.5×
• "Together" or "sum" = add the ages.
• "Older than" = +, "Younger than" = -
• Draw a table if it helps: Now, Past, Future.

Shortcut: When you see "in N years," always add N. When you see "N years ago," always subtract N. This never changes.

More Practice Examples

Example 1: In 6 years, Tom will be 20. How old now?
Step 1: x + 6 = 20
Step 2: x = 14
Answer: 14 years old

Example 2: 4 years ago, Sarah was 10. How old now?
Step 1: x - 4 = 10
Step 2: x = 14
Answer: 14 years old

Example 3: Peter is 3 times Mary's age. Together = 24. How old each?
Step 1: Mary = x, Peter = 3x
Step 2: x + 3x = 24
Step 3: 4x = 24 → x = 6
Answer: Mary = 6, Peter = 18

Example 4: Dad is 35, son is 7. In how many years will Dad be 3 times son?
Step 1: Let x = years
Step 2: Dad = 35 + x, Son = 7 + x
Step 3: 35 + x = 3(7 + x)
Step 4: 35 + x = 21 + 3x
Step 5: 35 - 21 = 3x - x → 14 = 2x → x = 7
Answer: In 7 years

Example 5: Anna is 15. Ben is 5 years younger. How old will Ben be in 8 years?
Step 1: Ben now = 15 - 5 = 10
Step 2: In 8 years = 10 + 8 = 18
Answer: 18 years old

What To Remember

• Age problems use "now," "before" (past), and "after" (future).
• Start with "now" as your variable (x).
• Past = x - years, Future = x + years.
• Age difference stays the same forever.
• All people age by the same amount.
• "Twice" = 2×, "Three times" = 3×, "Half" = 0.5×
• "Together" = add, "Older than" = +, "Younger than" = -
• Write equation, solve, then check.

HOW TO CREATE AN EQUATION IN AGE PROBLEMS (Step-by-Step Guide)

Many people get confused about creating equations. This guide will show you EXACTLY how to do it. I will explain: (1) when to use x, (2) where to put =, (3) which signs to use (+, -, ×), (4) where to put everything, and (5) how to check if your equation is correct.

PART 1: When to Use x (The Variable)

x is used for the UNKNOWN age that you need to find.

• If the question asks "How old is John now?" → let x = John's age now
• If the question asks "How old is Mary?" → let x = Mary's age
• If there are 2 people and you need to find both → start with the smaller/younger person as x

Golden Rule: Always start with "now" as x. Past = x - years. Future = x + years.

Example: "In 5 years, Anna will be 20. How old is she now?"

Step 1: What is unknown? → Anna's age now
Step 2: Write → Let x = Anna's age now

PART 2: How to Write Past and Future in Terms of x

This is the MOST IMPORTANT part. You must write ages in the past and future using x.

Rule for PAST (ago, before):

If someone was N years ago, write: x - N

Example: 5 years ago → x - 5

Rule for FUTURE (in, will be):

If someone will be N years in the future, write: x + N

Example: In 7 years → x + 7

Rule for NOW:

Current age is just: x

Example 1: "3 years ago, Ben was 12. How old is he now?"

Let x = Ben's age now
3 years ago = x - 3
Equation: x - 3 = 12

Example 2: "In 8 years, Maria will be 25. How old is she now?"

Let x = Maria's age now
In 8 years = x + 8
Equation: x + 8 = 25

PART 3: How to Handle TWO People

When there are 2 people, you need to express BOTH ages. Start with one as x, then express the other in terms of x.

Key Words and What They Mean:

• "Twice as old" → × 2 (or 2x)
• "Three times as old" → × 3 (or 3x)
• "Older than by N" → + N
• "Younger than by N" → - N
• "Together" or "sum" → add both ages

Example 1: "John is twice as old as Mary. Together, their ages = 30."

Step 1: Let x = Mary's age (younger person)
Step 2: John = twice Mary → John = 2x
Step 3: Together = 30 → x + 2x = 30
Equation: x + 2x = 30

Example 2: "Alice is 8 years older than Bob. Alice is 25."

Step 1: Let x = Bob's age
Step 2: Alice = 8 years older → Alice = x + 8
Step 3: Alice = 25 → x + 8 = 25
Equation: x + 8 = 25

PART 4: How to Put the "=" Sign

The "=" sign means "equals" or "is the same as." You put it where the problem says two things are equal.

Look for these words to find where to put "=":
- "is" → put = before the number
- "will be" → put = before the future number
- "was" → put = before the past number
- "equals" → put = right there
- "together is" → add both, then put =

Example 1: "In 5 years, Anna will be 20."

Anna in 5 years = x + 5
"will be 20" → = 20
Equation: x + 5 = 20

Example 2: "John is twice as old as Mary. Together = 30."

Mary = x
John = 2x
Together = x + 2x
"Together = 30" → x + 2x = 30
Equation: x + 2x = 30

PART 5: Two People at DIFFERENT Times (HARD TYPE)

This is the hardest type. You need to write BOTH people's ages at BOTH times.

Method: Write a table!

Person	Now	In N years
Person A	x	x + N
Person B	expression	expression + N

Example: "Father is 3 times as old as son. In 5 years, Father will be 2 times son. How old now?"

Step 1: Let x = son's age now
Step 2: Father = 3x (now)
Step 3: In 5 years: Son = x + 5, Father = 3x + 5
Step 4: "Father will be 2 times son" → 3x + 5 = 2(x + 5)

Equation: 3x + 5 = 2(x + 5)

PART 6: How to Check If Your Equation Is Correct

After you write an equation, solve it and then CHECK if the answer makes sense.

Check Steps:

• Step 1: Solve the equation to get x
• Step 2: Plug x back into the problem
• Step 3: Check if the numbers match what the problem says

Example: "In 5 years, Anna will be 20." Equation: x + 5 = 20

Step 1: Solve → x = 15
Step 2: Anna is 15 now. In 5 years = 15 + 5 = 20
Step 3: Problem says "will be 20" → YES, matches!

Answer is correct.

PART 7: FINAL CHECKLIST Before Solving

Before you solve, ask yourself:

• ✓ Did I let x = the unknown age "now"?
• ✓ Did I write past as (x - years) and future as (x + years)?
• ✓ For 2 people, did I express both in terms of x?
• ✓ Did I put "=" where the problem says "is," "was," "will be," or "equals"?
• ✓ For 2 people at different times, did I write both ages at both times?

If you answer YES to all 5 questions, your equation is correct!

Remember these:

• Age Problem - Problem about people's ages at different times.
• Now - Current age (use as variable x).
• Past / Ago - Subtract years (x - years).
• Future / In / Will be - Add years (x + years).
• Age Difference Stays Same - The difference between ages never changes.
• All Age Same Amount - Everyone gets older by the same number of years.
• Twice - 2 times (2×).
• Three Times - 3 times (3×).
• Half - 0.5 times (0.5×).
• Together / Sum - Add the ages.
• Older Than - Add (+).
• Younger Than - Subtract (-).
• Variable - Use x for unknown age.

Multiple Choice Questions

Answer these in the comment section. Choose the best answer for each item.

1. In 5 years, Lisa will be 22. How old is she now?

A. 15
B. 16
C. 17
D. 18

2. 3 years ago, Mark was 12. How old is he now?

A. 14
B. 15
C. 16
D. 17

3. John is twice as old as Mary. Together, their ages = 36. How old is John?

A. 12
B. 18
C. 24
D. 30

4. Dad is 40, son is 10. In how many years will Dad be 2 times son?

A. 10
B. 15
C. 20
D. 25

5. Anna is 18. Ben is 6 years younger. How old will Ben be in 5 years?

A. 15
B. 16
C. 17
D. 18