Important Formulas on Problems on Ages
1. If the current age is ‘x’, then ‘n’ times the age is ‘nx’.
2. If the current age is x, then age n years later/hence = (x + n).
3. If the current age is ‘x’, then age ‘n’ years ago = (x – n).
4. The ages in a ratio ‘a : b’ will be ‘ax’ and ‘bx’.
5. If the current age is x, then 1/n of the age is x/n.
Example. 01
A
father was 4 times as old as his son 8 years ago. Eight years hence,
father will be twice as old as his son. Find their present ages.
father was 4 times as old as his son 8 years ago. Eight years hence,
father will be twice as old as his son. Find their present ages.
Solution:
Let son’s age 8 years ago be x years.
Thus, father’s age at that time = 4x years
After 8 years, son’s age = (x + 8) + 8 = (x+16) years
After 8 years, father’s age = (4x + 8) + 8 = (4x+16) years
So, According to Question, 2(x + 16) = 4x + 16 or x = 8
Therefore, The present age of the son = x + 8 = 16 years
The present age of the father = 4x + 8 = 32 + 8= 40 years
Example. 02
Father
is aged three times more than his son. After 8 years, he would be two
and a half times of his son’s age. After further 8 years, how many times
would he be of his son’s age?
is aged three times more than his son. After 8 years, he would be two
and a half times of his son’s age. After further 8 years, how many times
would he be of his son’s age?
Solution:
Let Son’s present age be ‘x’ years.
Then, father’s present age =(x + 3x) years = 4x years.
Therefore, (4x + 8) = (2/5) (x + 8)
=> 8x + 16 = 5x + 40
=> 3x = 24 => x = 8.
After further 8 years, Son’s age will be (x + 16) = 24 years.
And father’s age will be (4x +16) = 48 years.
Hence, the required ratio is (4x +16)/(x+16) = 48/24 = 2.
Example. 03
A man is 24
years older than his son. In two years, his age will be twice the age of his
son. The present age of his son is:
years older than his son. In two years, his age will be twice the age of his
son. The present age of his son is:
(A) 14 years
(B) 18 years
(C) 20 years
(D) 22 years
Answer:
Option D
Option D
Solution:
Let the
son’s present age be x years. Then, man’s present age = (x +
24) years.
son’s present age be x years. Then, man’s present age = (x +
24) years.
∴ (x +
24) + 2 = 2(x + 2)
24) + 2 = 2(x + 2)
=> x + 26 = 2x +
4
=> x = 22.
4
=> x = 22.
SOLVED EXAMPLE SET 1
Question
No. 01
No. 01
A
father was 4 times as old as his son 8 years ago. Eight years hence, father
will be twice as old as his son. Find their present ages.
father was 4 times as old as his son 8 years ago. Eight years hence, father
will be twice as old as his son. Find their present ages.
Solution:
Let son’s age 8 years ago be x years.
Thus, father’s age at that time = 4x years
After 8 years, son’s age = (x + 8) + 8 = (x+16) years
After 8 years, father’s age = (4x + 8) + 8 = (4x+16) years
So, According to Question, 2(x + 16) = 4x + 16 or x = 8
Therefore, The present age of the son = x + 8 = 16 years
The present age of the
father = 4x + 8 = 32 + 8= 40
years
father = 4x + 8 = 32 + 8= 40
years
Question
No. 02
No. 02
Father
is aged three times more than his son. After 8 years, he would be two and a
half times of his son’s age. After further 8 years, how many times would he be
of his son’s age?
is aged three times more than his son. After 8 years, he would be two and a
half times of his son’s age. After further 8 years, how many times would he be
of his son’s age?
Solution:
Let Son’s present age be ‘x’ years.
Then, father’s present age =(x + 3x)
years = 4x years.
years = 4x years.
Therefore, (4x + 8) = (2/5) (x +
8)
8)
=> 8x + 16 = 5x + 40
=> 3x = 24 => x = 8.
After further 8 years, Son’s age will be
(x + 16) = 24 years.
(x + 16) = 24 years.
And father’s age will be (4x +16) = 48 years.
Hence, the required ratio is (4x
+16)/(x+16) = 48/24 = 2.
+16)/(x+16) = 48/24 = 2.
Question
No. 03
No. 03
The
sum of ages of 5 children born at the intervals of 3 years each is 50 years.
What is the age of the youngest child?
sum of ages of 5 children born at the intervals of 3 years each is 50 years.
What is the age of the youngest child?
Solution:
Let the ages of children be x, (x
+ 3), (x + 6), (x + 9) and (x + 12) years.
+ 3), (x + 6), (x + 9) and (x + 12) years.
Then, x + (x + 3) + (x +
6) + (x + 9) + (x + 12) = 50
6) + (x + 9) + (x + 12) = 50
=> 5x = 20
=> x = 4.
Age of the youngest child = x = 4
years.
years.
Question
No. 04
No. 04
‘A’
is twice as old as ‘B’ was two years ago. If the difference in their ages be 2
years, find A’s age.
is twice as old as ‘B’ was two years ago. If the difference in their ages be 2
years, find A’s age.
Solution:
Let B’s age 2 years ago be x years
Therefore, A’s present age = 2x years
Also 2x – (x + 2) = 2 or x=4
So, A’s age = 2x = 2 × 4 = 8 years
Question
No. 05
No. 05
A
father said to his son, “I was as old as you are at the present at the
time of your birth”. If the father’s age is 38 years now, the son’s age
five years back was:
father said to his son, “I was as old as you are at the present at the
time of your birth”. If the father’s age is 38 years now, the son’s age
five years back was:
Solution:
Let the son’s present age be x years.
Then, (38 – x) = x
Then, (38 – x) = x
=> 2x = 38
=> x = 19.
Son’s age 5 years back (19 – 5) = 14
years.
years.
Question
No. 06
No. 06
A
is two years older than B who is twice as old as C. If the total of the ages of
A, B and C be 27, the how old is B?
is two years older than B who is twice as old as C. If the total of the ages of
A, B and C be 27, the how old is B?
Solution:
Let C’s age be x years.
Then, B’s age = 2x years. A’s age
= (2x + 2) years.
= (2x + 2) years.
Therefore, (2x + 2) + 2x + x
= 27
= 27
=> 5x = 25
=> x = 5.
Hence, B’s age = 2x = 10 years.
Question
No. 07
No. 07
Present
ages of ‘A’ and ‘B’ are in the ratio of 5 : 4 respectively. Three years hence,
the ratio of their ages will become 11 : 9 respectively. What is B’s present
age in years?
ages of ‘A’ and ‘B’ are in the ratio of 5 : 4 respectively. Three years hence,
the ratio of their ages will become 11 : 9 respectively. What is B’s present
age in years?
Solution:
Let the present ages of ‘A’ and ‘B’ be 5x
years and 4x years respectively.
years and 4x years respectively.
Then, (5x + 3)/ (4x + 3) = 11/9
=> 9(5x + 3) = 11(4x +
3)
3)
=> 45x + 27 = 44x + 33
=> 45x – 44x = 33 – 27
=> x = 6.
B’s present age = 4x = 24 years.
Question
No. 08
No. 08
The
age of a father 10 years ago was thrice the age of his son. Ten years hence,
the father’s age will be twice that of his son. What is the ratio of their
present ages?
age of a father 10 years ago was thrice the age of his son. Ten years hence,
the father’s age will be twice that of his son. What is the ratio of their
present ages?
Solution:
Let the present ages of father and son be
‘x’ and ‘y’ years respectively.
‘x’ and ‘y’ years respectively.
Then (x – 10) = 3 (y – 10) or 3y – x = 20 —— (1)
And (x+10) = 2 (y + 10) or x –
2y = 10 ——- (2)
2y = 10 ——- (2)
Adding eq. (1) + (2) => y
= 30
= 30
Substituting the value of y = 30 in
equation (1) we get x = 70
equation (1) we get x = 70
Ratio of their ages = 70: 30 or 7:3
Question
No. 09
No. 09
A
man is 24 years older than his son. In two years, his age will be twice the age
of his son. The present age of his son is:
man is 24 years older than his son. In two years, his age will be twice the age
of his son. The present age of his son is:
Solution:
Let the son’s present age be x years.
Then, man’s present age = (x + 24)
years.
years.
Therefore, (x + 24) + 2 = 2(x +
2)
2)
=> x + 26 = 2x + 4
=> x = 22.
Hence, the present age of his son is 22
years.
years.
Question
No. 10
No. 10
Six
years ago, the ratio of the ages of ‘A’ and ‘B’ was 6 : 5. Four years hence,
the ratio of their ages will be 11 : 10. What is B’s age at present?
years ago, the ratio of the ages of ‘A’ and ‘B’ was 6 : 5. Four years hence,
the ratio of their ages will be 11 : 10. What is B’s age at present?
Solution:
Let the ages of ‘A’ and ‘B’ 6 years ago
be 6x and 5x years respectively.
be 6x and 5x years respectively.
Then, [(6x + 6) + 4[/ [(5x +
6) + 4] =11/10
6) + 4] =11/10
=> 10(6x + 10) = 11(5x +
10)
10)
=> 5x = 10
=> x = 2.
Therefore, B’s present age = (5x +
6) = 16 years.
6) = 16 years.
SOLVED EXAMPLE SET 2
Question
No. 01
No. 01
Ratio
of A’s age to B’s age is equal to 4:3. A will be 26 years old after 6 years.
How old is B now?
of A’s age to B’s age is equal to 4:3. A will be 26 years old after 6 years.
How old is B now?
Solution:
A’s present age = (26 – 6) = 20 years
B’s present age = 20 × (3/4) = 15 years
Question
No. 02
No. 02
The
sum of the present ages of a father and his son is 60 years. Six years ago,
father’s age was five times the age of the son. After 6 years, son’s age will
be:
sum of the present ages of a father and his son is 60 years. Six years ago,
father’s age was five times the age of the son. After 6 years, son’s age will
be:
Solution:
Let the present ages of son and father be
‘x’ and (60 –x) years respectively.
‘x’ and (60 –x) years respectively.
Then, (60 – x) – 6 = 5(x –
6)
6)
=> 54 – x = 5x – 30
=> 6x = 84
=> x = 14.
Son’s age after 6 years = (x+ 6) =
20 years.
20 years.
Question
No. 03
No. 03
At present, the ratio between the ages of ‘A’
and ‘B’ is 4 : 3. After 6 years, A’s age will be 26 years. What is the age of ‘B’
at present?
and ‘B’ is 4 : 3. After 6 years, A’s age will be 26 years. What is the age of ‘B’
at present?
Solution:
Let the present ages of ‘A’ and ‘B’ be ‘4x’
years and ‘3x’ years respectively.
years and ‘3x’ years respectively.
Then, 4x + 6 = 26
=> 4x = 20
=> x = 5.
B’s age = 3x = 15 years.
Question
No. 04
No. 04
A
is younger than B by 7 years. If their ages are in the respective ratio of 7 :
9, how old is A?
is younger than B by 7 years. If their ages are in the respective ratio of 7 :
9, how old is A?
Solution:
Let B’s age be ‘x’ years.
Then, A’s age = (x – 7) years.
=> (x – 7)/x
=7/9
=7/9
=> (9x – 63) = 7x
=> 2x = 63
=> x = 31.5
Hence, A’s age =(x – 7) = 24.5
years.
years.
Question
No. 05
No. 05
The
ratio of the ages of father and son at present is 6:1. After 5 years the ratio
will become 7:2. The present age of the son is:
ratio of the ages of father and son at present is 6:1. After 5 years the ratio
will become 7:2. The present age of the son is:
Solution:
Let their present ages be 6x and x years respectively.
Then 6x + 5)/(x + 5) = 7/2 = 2 (6x + 5) = 7 (x + 5) or, x=5
Therefore, Present age of the son = 5
years.
years.
Question
No. 06
No. 06
The
present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the
sum of their ages was 56. Find their present ages (in years).
present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the
sum of their ages was 56. Find their present ages (in years).
Solution:
Let their present ages be 4x, 7x
and 9x years respectively.
and 9x years respectively.
Then, (4x – 8) + (7x – 8) +
(9x – 8) = 56
(9x – 8) = 56
=> 20x = 80
=> x = 4.
Their present ages are 4x = 16
years, 7x = 28 years and 9x = 36 years respectively.
years, 7x = 28 years and 9x = 36 years respectively.
Question
No. 07
No. 07
Alisha’s
father was 38 years of age when she was born while her mother was 36 years old
when her brother, four years younger to her was born. What is the difference
between the ages of her parents?
father was 38 years of age when she was born while her mother was 36 years old
when her brother, four years younger to her was born. What is the difference
between the ages of her parents?
Solution:
Mother’s age when Alisha’s brother was
born = 36 years.
born = 36 years.
Father’s age when Alisha’s brother was
born = (38 + 4) years = 42 years.
born = (38 + 4) years = 42 years.
Required difference = (42 – 36) years = 6
years.
years.
Question
No. 08
No. 08
A
person’s present age is two-fifth of the age of his mother. After 8 years, he
will be one-half of the age of his mother. How old is the mother at present?
person’s present age is two-fifth of the age of his mother. After 8 years, he
will be one-half of the age of his mother. How old is the mother at present?
Solution:
Let, the mother’s present age be ‘x’ years.
Then, the person’s present age = [(2/5) x] years.
Therefore, [(2/5)(x + 8)] = [(1/2)(x +
8)]
8)]
=> 2(2x + 40) = 5(x + 8)
=>x = 40.
Hence, at present mother’s age is 40
years.
years.
Question
No. 09
No. 09
Three
years ago the average age of ‘A’ and ‘B’ was 18 years. With ‘C’ joining them
now, the average becomes 22 years. How old is C now?
years ago the average age of ‘A’ and ‘B’ was 18 years. With ‘C’ joining them
now, the average becomes 22 years. How old is C now?
Solution:
(A+B)’s total present age = (2 x 18+3+3) =
42 years —– (1)
42 years —– (1)
(A+B+C)’s total present age = 22 x 3 = 66
years ——– (2)
years ——– (2)
Substituting Eq. (2) from (1), C’s age =
66-42 = 24 years
66-42 = 24 years
Question
No. 10
No. 10
The
age of father 10 years ago was thrice the age of his son. Ten years hence,
father’s age will be twice that of his son. The ratio of their present ages is:
age of father 10 years ago was thrice the age of his son. Ten years hence,
father’s age will be twice that of his son. The ratio of their present ages is:
Solution:
Let the ages of father and son 10 years
ago be ‘3x’ and ‘x’ years respectively.
ago be ‘3x’ and ‘x’ years respectively.
Then, (3x + 10) + 10 = 2[(x +
10) + 10]
10) + 10]
=> 3x + 20 = 2x + 40
=> x = 20.
Required ratio = (3x + 10) : (x
+ 10) = 70 : 30 = 7 : 3.
+ 10) = 70 : 30 = 7 : 3.