Problem on Ages

UTKALPRESS

Problems on Ages

Arithmetic Book Note • Concepts • Formulae • Solved Examples

1. Introduction

Problems on Ages are arithmetic questions in which the present, past or future ages of two or more persons are related through equations, ratios or differences. The most important idea is that although the actual ages change with time, the difference between two persons' ages remains constant.

2. Basic Age Principle

Basic Rule:

Future Age = Present Age + Number of Years

Past Age = Present Age − Number of Years

Age Difference = Older Age − Younger Age

Example: Ravi is 25 years old. What will be his age after 8 years?

Future age = 25 + 8
= 33 years.

3. Age Difference

If A is 8 years older than B today, A will still be 8 years older than B after 5 years and also 8 years older 10 years ago.

Important Formula:

Age difference = Constant

For two persons:
A − B = Constant

Example: Father is 35 years older than his son. After 10 years, what will be the difference between their ages?

The difference remains unchanged.
Answer = 35 years.

4. Present, Past and Future Age Table

Time	A's Age	B's Age
10 years ago	A − 10	B − 10
5 years ago	A − 5	B − 5
Present	A	B
5 years later	A + 5	B + 5
10 years later	A + 10	B + 10

5. Age Ratio

If present ages of A and B are in the ratio:

A : B = m : n

Then their ages can be written as:

A = mx, B = nx

Example: The ages of A and B are in the ratio 3 : 5. If their total age is 40, find their ages.

3x + 5x = 40
8x = 40
x = 5

A = 3 × 5 = 15 years
B = 5 × 5 = 25 years

UTKALPRESS

Problems on Ages

Ratio Method • Equation Method • Past & Future Ages

6. Ratio of Ages After Some Years

Suppose present ages are:

A = ax
B = bx

After T years:
A + T = ax + T
B + T = bx + T

Therefore:
(A+T)/(B+T) = required ratio

Example: The present ages of A and B are in the ratio 4 : 5. After 6 years, their ages will be in the ratio 5 : 6. Find their present ages.

Let ages = 4x and 5x.

After 6 years:
(4x+6)/(5x+6) = 5/6

6(4x+6) = 5(5x+6)
24x + 36 = 25x + 30
x = 6

A = 4 × 6 = 24 years
B = 5 × 6 = 30 years

7. Ratio in the Past

If present ages are A and B and the ratio T years ago was m:n:

(A−T)/(B−T) = m/n

Example: Five years ago, the ages of A and B were in the ratio 3 : 4. Their present age difference is 5 years. Find their present ages.

Let their ages five years ago be 3x and 4x.

Difference = 4x − 3x = x = 5
Therefore x = 5.

Five years ago:
A = 15, B = 20

Present:
A = 15 + 5 = 20 years
B = 20 + 5 = 25 years.

8. Sum of Ages

If there are n persons and the sum of their present ages is S:

Average age = S/n

After T years:
New sum = S + nT

Example: The total age of 4 friends is 100 years. What will be their total age after 5 years?

Increase = 4 × 5 = 20
New total = 100 + 20
= 120 years.

9. Average Age

Average Age = Total Age / Number of Persons

If each of n persons becomes T years older:
New Average = Old Average + T

Example: The average age of 5 students is 16 years. What will be their average age after 4 years?

New average = 16 + 4
= 20 years.

10. Equation Method

When the wording becomes complicated, assign a variable to the unknown age and translate every statement into an equation.

For example, if B is 7 years younger than A:
A = B + 7

If A is twice B:
A = 2B

Remember: Never change the age difference when moving from present to past or future. Both persons move by the same number of years.

UTKALPRESS

Problems on Ages

Special Cases • Parents & Children • Ratio Changes

11. Parent and Child Problems

In parent-child questions, the age difference remains constant. However, the ratio of their ages changes with time. This is the main source of many questions.

If father's present age = F and son's present age = S:

Age difference = F − S

After T years:
Father = F + T
Son = S + T

Ratio after T years:
(F+T):(S+T)

Example: A father is 4 times as old as his son. After 8 years, he will be twice as old as his son. Find their present ages.

Let son's age = x.
Father's age = 4x.

After 8 years:
4x + 8 = 2(x + 8)
4x + 8 = 2x + 16
2x = 8
x = 4

Son = 4 years
Father = 16 years.

12. When Ratio Becomes Known

If two present ages have ratio a:b and their age difference is D:

(a−b)x = D

Therefore:
x = D/(a−b)

Ages:
A = ax
B = bx

Example: The ages of two persons are in the ratio 7 : 5. Their difference is 10 years. Find their ages.

(7 − 5)x = 10
2x = 10
x = 5

A = 7 × 5 = 35 years
B = 5 × 5 = 25 years.

13. Age Ratio Changes — Shortcut

Suppose present ages are ax and bx. If after T years the ratio becomes p:q, use:

(ax+T)/(bx+T) = p/q

Cross multiply directly. This is usually the fastest method in competitive examinations.

14. Birth Year Formula

Birth Year:

Birth Year = Current Year − Present Age

For a person aged A in year Y:
Birth Year ≈ Y − A

If an exact birthday has not yet occurred in the current year, the exact birth date/year interpretation may differ by one year. For ordinary aptitude questions, follow the information explicitly given in the question.

15. Three-Person Age Problems

If ages are in ratio:
a : b : c

and total age is S:
x = S/(a+b+c)

Then:
A=ax, B=bx, C=cx

Example: The ages of A, B and C are in the ratio 2 : 3 : 5. Their total age is 50 years.

Total ratio = 2 + 3 + 5 = 10
x = 50/10 = 5

A = 10 years
B = 15 years
C = 25 years.

16. Important Age Formula Table

Situation	Formula
Future age	Present + T
Past age	Present − T
Age difference	Older − Younger
Total age	Sum of individual ages
Average age	Total age ÷ Number of persons
Ratio ages	ax : bx
Ratio after T years	(ax+T):(bx+T)

UTKALPRESS

Problems on Ages

Quick Revision • Solved Examples • Practice Set

17. Quick Age Shortcuts

Question Type	Quick Method
Position after T years	Add T to every person's age.
Age T years ago	Subtract T from every person's age.
Age difference	Subtract the smaller age from the larger.
Ratio + total	Divide total by sum of ratio terms.
Ratio + difference	Difference ÷ difference of ratio terms.
Future ratio	Add T to both ages, then form ratio.
Past ratio	Subtract T from both ages, then form ratio.
Average age	Total age ÷ number of persons.

18. Solved Example — Future Ratio

Question: The present ages of A and B are in the ratio 5 : 7. After 6 years, the ratio becomes 3 : 4. Find their present ages.

Let ages = 5x and 7x.

(5x+6)/(7x+6) = 3/4

20x + 24 = 21x + 18
x = 6

A = 5 × 6 = 30 years
B = 7 × 6 = 42 years.

19. Solved Example — Past Ratio

Question: Eight years ago, the ages of A and B were in the ratio 3 : 5. Their present age difference is 12 years. Find their present ages.

Let their ages 8 years ago be 3x and 5x.

Difference = 5x − 3x = 2x
2x = 12
x = 6

Eight years ago:
A = 18, B = 30

Present:
A = 26 years
B = 38 years.

Answer: 26 years and 38 years.

20. Practice Questions

1. A man is 32 years old. What will be his age after 7 years?

2. A woman is 45 years old. What was her age 9 years ago?

3. The ages of A and B are in the ratio 3 : 5 and their sum is 40 years. Find their ages.

4. A is 12 years older than B. If B is 20 years old, find A's age.

5. The ages of two persons are in the ratio 4 : 7. Their difference is 15 years. Find their ages.

6. The average age of 6 students is 15 years. What will be their average age after 5 years?

7. The total age of 5 persons is 120 years. What will be their total age after 4 years?

8. A father is 3 times as old as his son. Their total age is 48 years. Find their ages.

9. Five years ago, the ages of A and B were in the ratio 2 : 3. Their present age difference is 8 years. Find their present ages.

10. The ages of A, B and C are in the ratio 2 : 3 : 4. If their total age is 63 years, find their ages.

21. Answer Key

Q.	Answer	Q.	Answer
1	39 years	6	20 years
2	36 years	7	140 years
3	15, 25 years	8	12, 36 years
4	32 years	9	21, 29 years
5	20, 35 years	10	14, 21, 28 years

22. Final Revision Notes

Remember these 5 rules:

1. Age increases by exactly the number of years passed.
2. Age decreases by exactly the number of years gone back.
3. Age difference between two people is always constant.
4. Age ratios change with time.
5. In ratio problems, use ax, bx, cx and form an equation.

Golden Rule:
Whenever you see a problem on ages, first decide whether the question deals with present age, past age, future age, ratio, difference, total or average. Once the correct relationship is written, the problem usually becomes a simple equation.