Problems on Ages: Formulas, Tricks and Solved Questions

Problems on Ages use present, past or future ages to form equations involving sums, differences and ratios. The age difference between two people remains constant, while their age ratio changes with time. This page explains standard age formulas, equation-based methods, shortcuts and solved examples for quantitative aptitude questions.

On this page

What Are Problems on Ages?

Problems on Ages are arithmetic questions in which the ages of one or more people are related through time, difference, sum or ratio. They are solved by representing an unknown age with a variable and forming an equation.

If a person's present age is x years, the age after n years is x + n, and the age n years ago was x - n. For two people, their age difference remains unchanged. For example, if A is 5 years older than B and B is 20 years old, A is 25 years old now and the difference remains 5 years in every time period.

Problems on Ages Formula & Tricks

Important Formulas

Age after n years
Future age = Present age + n

Add the number of elapsed years to the current age. If a person is 18 now, the age after 7 years is 18 + 7 = 25 years.

Age n years ago
Past age = Present age - n

Subtract the elapsed years from the current age. A person aged 30 now was 30 - 8 = 22 years old 8 years ago.

Constant age difference
Age difference = Older age - Younger age

The difference between two ages is the same in the past, present and future. If the present ages are 36 and 24, the difference is 12 years in every period.

Age ratio
Older age / Younger age = Required age ratio

For a ratio a:b, write the ages as ax and bx. If their sum is S, then ax + bx = S, so x = S/(a + b).

Average age
Average age = Sum of ages / Number of people

The total age is obtained by multiplying average age by the number of people. If the average age of 4 people is 21 years, their total age is 4 × 21 = 84 years.

Quick Tricks

Keep the age difference unchanged

When the same number of years is added to or subtracted from two ages, their difference does not change. Use this directly before forming longer equations.

Example: If a father is 28 years older than his son, the difference is 28 years both 6 years ago and 10 years later.
Use ratio parts for present ages

If two present ages are in the ratio a:b, represent them as ax and bx. This converts the ratio into a simple linear equation.

Example: For ages in the ratio 3:5 with a total of 64, let the ages be 3x and 5x. Then 8x = 64, so the ages are 24 and 40.
Convert future or past ratios carefully

Add or subtract the time from each person's age before applying the ratio. Do not apply the ratio directly to the present ages unless the question states a present-age ratio.

Example: If a father and son will be in the ratio 3:1 after 8 years, write their future ages as F + 8 and S + 8, then use (F + 8)/(S + 8) = 3.

Problems on Ages Concepts

Representing Ages with Variables

An unknown present age is commonly represented by x, and related ages are written by adding or subtracting the stated difference.

If Riya is x years old, her brother who is 4 years older is x + 4, while a sister who is 3 years younger is x - 3. For n years, use x + n for the future and x - n for the past.

Example: If the sum of a person's present age and age after 6 years is 34, then x + (x + 6) = 34. Therefore, 2x = 28 and x = 14 years.

Age Problems Based on Sum and Difference

When the sum and difference of two ages are known, the older age is half their sum plus half their difference, and the younger age is half their sum minus half their difference.

For total age S and age difference D: older age = (S + D)/2 and younger age = (S - D)/2. Both ages must be non-negative and the sum and difference should have the same parity for whole-number ages.

Example: If two people's total age is fifty-six years and their difference is 12 years, older age = (56 + 12)/2 = 34 years and younger age = (56 - 12)/2 = 22 years.

Age Ratio Equations

If two ages are in the ratio a:b, write them as ax and bx, where x is the common factor.

If their present total is S, then (a + b)x = S. If their difference is D, then (b - a)x = D, assuming b is the larger ratio part. For a ratio at a different time, add or subtract the time from both ages before applying the ratio.

Example: The present ages of A and B are in the ratio 4:7 and their difference is 15 years. Thus, (7 - 4)x = 15, so x = 5. Their ages are 20 and 35 years.

Past and Future Age Conditions

Past and future conditions are formed by shifting every person's present age by the same number of years.

If present ages are x and y, their ages after n years are x + n and y + n; their ages n years ago are x - n and y - n. A condition such as a future ratio is written as (x + n)/(y + n), not x/y.

Example: A father is three times his son's age now. If the son is 10 years old, the father is 30. After 5 years, their ages are 35 and 15, giving a ratio of 35:15 = 7:3.

Average Age and Change in Group

The total age of a group equals its average age multiplied by the number of people.

When a person joins or leaves a group, first calculate the old and new total ages, then find the age of the entering or leaving person by subtraction. If every member becomes one year older, the average also increases by one year.

Example: The average age of 5 people is 24 years, so their total age is 120 years. When a sixth person joins, the average becomes 25 years and the new total is 150 years. The new person's age is 150 - 120 = 30 years.

Problems on Ages Video Lessons

Watch short topic-wise lessons for quick revision.

3 Lessons
Lesson 1 of 3 Quick Revision

Age Problems: Present, Past and Future

Learn how to set up age problems using present, past and future age relationships, creating the correct equations before solving quantitative aptitude questions.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Problems on Ages Lessons Scroll to explore →

Practice Problems on Ages Questions

Practise published questions related to this topic.

1A vendor bought 100 oranges at ₹8 each. He sold 80 of them at ₹10 each and the remaining 20 oranges at ₹6 each. What is his overall profit or loss percentage?→ 2An article is sold at a loss of 12% on its cost price. If the same article were sold for ₹84 more than its original selling price, the seller would make a profit of 8% on the same cost price. What is the cost price of the article?→ 3Simplify: ((3² × 9)³)/(27² × 3⁴)→ 4Least Common Multiple of A and B is 20. Least Common Multiple of C and D is 30. What is the Least Common Multiple of A, B, C and D?→ 5Maya buys a mobile phone for her child. The phone is marked at ₹21,000 and is offered with two successive discounts of 5% and Z%, respectively. After receiving both discounts, she pays ₹11,172. Find the value of Z.→ 6Total marked price of 600 articles is Rs. 450. 40 percent discount is given. What will be the selling price for 18000 such articles? (marked price of each article is same)→ 7A dealer purchased an AC for ₹1,848. After allowing a discount of 58% on its marked price, he still gains 63%. Find the marked price of the AC.→ 8If we multiply (-6ab) and (11ab²c), then we get:→ 9Find the HCF of the numbers 2520, 3600, 5880.→ 10What is the least number that should be added to 517 so that the number obtained is exactly divisible by 15, 25 and 8?→

Problems on Ages Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Problems on Ages: Quick Revision

Use these rules to translate age statements into equations.

  • Age after n years = Present age + n.
  • Age n years ago = Present age - n.
  • The age difference between two people remains constant.
  • For ratio a:b, represent the ages as ax and bx.
  • If the sum is S and difference is D, older age = (S + D)/2 and younger age = (S - D)/2.
  • For a future or past ratio, add or subtract the time from both ages before forming the ratio.
  • Total age = Average age × Number of people.

Problems on Ages FAQs

What is the formula for a person's age after n years?

Future age = Present age + n. For example, a person aged 27 now will be 27 + 6 = 33 years old after 6 years.

How do you calculate age n years ago?

Past age = Present age - n. A person aged 32 now was 32 - 9 = 23 years old 9 years ago.

Does the age difference between two people change with time?

No. If two people differ by 18 years now, their difference was 18 years in the past and will remain 18 years in the future.

Two ages have a sum of 70 and a difference of 16. What are the ages?

Older age = (70 + 16)/2 = 43 years. Younger age = (70 - 16)/2 = 27 years.

How are ages represented when their ratio is 2:5?

Represent the ages as 2x and 5x. If their total is 49, then 7x = 49, so x = 7 and the ages are 14 and 35 years.

How should a future age ratio be written?

Add the future interval to both present ages first. If present ages are x and y, their ratio after n years is (x + n):(y + n).

Continue learning Problems on Ages on PrepShots

Continue on PrepShots