Present, Past & Future Ages

🎯 Learning Objective

Convert age statements into a linear equation using a single variable and solve for present age.

💡 Concept

• Let the present age be x. Then t years AGO the age was (x − t), and t years HENCE it will be (x + t).
• Every person in the problem ages by the SAME number of years over the same period.
• Turn each English phrase into algebra: 'is twice' → =2×, 'older by 5' → +5.
• Write one variable, form one equation from the given condition, and solve.
• 'Hence / after / later' means add; 'ago / back / earlier' means subtract.

🧮 Key Formulas

Age t years ago = x − t

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Age t years hence = x + t

✏️ Easy Example

Q. Rahul's present age is 25 years. What was his age 8 years ago, and what will it be 5 years hence?

1. 8 years ago = 25 − 8 = 17 years
2. 5 years hence = 25 + 5 = 30 years

Answer: 17 years ago, 30 years hence

🇮🇳 Real-Life Example

When you fill a railway recruitment form, the portal checks your age as on a cut-off date — it simply adds or subtracts years from your date of birth, exactly like these problems.

📝 Exam-Level Example

Q. The sum of the ages of a father and his son is 56 years. After 4 years, the father will be 3 times as old as the son. Find their present ages.

1. Let son = s, then father = 56 − s
2. After 4 years: (56 − s) + 4 = 3(s + 4)
3. 60 − s = 3s + 12
4. 48 = 4s → s = 12
5. Father = 56 − 12 = 44

Answer: Son = 12 years, Father = 44 years

📝 Exam-Level Example

Q. A man is 24 years older than his son. In 2 years, his age will be twice the age of his son. Find the present age of the son.

1. Let son = s, man = s + 24
2. In 2 years: (s + 24) + 2 = 2(s + 2)
3. s + 26 = 2s + 4
4. 26 − 4 = 2s − s → s = 22

Answer: Son = 22 years (man = 46 years)

🪄 Memory Trick

Read left to right and write algebra as you go: 'father' → F, 'is twice son' → F = 2S. The equation almost writes itself.

⚠️ Common Mistakes

• ❌ Adding years to only one person (everyone ages equally)
• ❌ Mixing up 'ago' (subtract) with 'hence' (add)
• ❌ Applying the multiplier to the present age instead of the future/past age

🏆 Exam Tips

• ✅ Assign the variable to the person whose age is asked or is smallest
• ✅ Always verify by plugging the answer back into the original words

📌 Summary

• t years ago → x − t; t years hence → x + t
• Everyone ages by the same t
• Translate words directly into one equation
• Check the answer against the sentence