Problems on Ages

Let the son's present age be x years.
Then, (38 - x) = x
2x = 38.
x = 19.
Son's age 5 years back = (19 - 5)
= 14 years.

Question: 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 42, then how old is B?

Solution:
Let the age of C be = x years 

 Then, age of B = 2x years.
age of A = (2x + 2) years.

According to the question,
(2x + 2) + 2x + x = 42
⇒ 5x + 2 = 42
⇒ 5x = 42 - 2
⇒ 5x = 40
⇒ x = 40/5
⇒ x = 8

Hence, age of B = 2x = (2 × 8) years = 16 years.

Let, after x years father's age will be twice
ATQ, 2(12 + x) = 40 + x
Or, 24 + 2x = 40 + x
Or, x = 16

Question: 6 years ago, the ratio of the ages of Kamrul and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present ?

Solution:
Let the ages of Kamrul and Sagar 6 years ago be 6x and 5x years
Then,
{(6x + 6) + 4}/{(5x + 6) + 4} = 11/10
⇒ (6x + 10)/(5x + 10) = 11/10
⇒ 10(6x + 10) = 11(5x + 10)
⇒ 60x + 100 = 55x + 110
⇒ 5x = 10
∴ x = 2

∴ Sagar's present age,
= (5x + 6) years
= (5 × 2 + 6) years
= 16 years

∴ Sagar's present age is 16 years. 

Sum of ages of A and B, 3 Years ago =(18 x 2) = 36 Years
Sum of ages of A, B, and C,
Now = (22 x 3) = 66 Years
Sum of ages of A and B,
Now = (36 +6)Years = 42 Years

Therefore,
C's age = (66 - 42)Years = 24 Years.

Question: The average age of the children in a tour group is 8 years, and that of the adults is 30 years. If the average age of the entire tour group is 15 years, find the ratio of children to adults in the group.

Solution:
Here,
Average age of children = 8 years
Average age of adults = 30 years
Average age of the entire group = 15 years

Let the number of children = m
and the number of adults = n

Then, the total number of people in the group is (m + n)

ATQ,
8m + 30n = 15(m + n)
⇒ 8m + 30n = 15m + 15n
⇒ 15m - 8m = 30n - 15n
⇒ 7m = 15n
⇒ m/n = 15/7
⇒ m : n = 15 : 7

∴ The ratio of children to adults in the group is 15 : 7

Question: Father is aged 3 times more than his son and after 8 years he will be 2.5 times, Now, what is the age of the son?

Solution:
মনে করি,
ছেলের বর্তমান বয়স x বছর
যদি বাবার বর্তমান বয়স ছেলের বর্তমান বয়স অপেক্ষা 3 গুণ বেশি হয়
তাহলে, বাবার বর্তমান বয়স = 3x বছর

8 বছর পর,
ছেলের বয়স হবে (x + 8) বছর
বাবার বয়স হবে (3x + 8) বছর

প্রশ্নমতে,
(3x + 8) = 2.5(x + 8)
⇒ (3x + 8) = 2.5x + 20
⇒ 3x - 2.5x = 20 - 8
⇒ 0.5x = 12
⇒ x = 12/0.5
∴ x = 24

∴ ছেলের বর্তমান বয়স 24 বছর।

Question: The average age of a group of 12 students is 18 years. When 3 more students join the group, the overall average age increases by 1 year. What is the average age of the three new students?

Solution:
Total age of 12 students = 12 × 18 = 216 years
After 3 new students joined, total students = 12 + 3 = 15
New average = 18 + 1 = 19 years
Total age of 15 students = 15 × 19 = 285 years
Total age of 3 new students = 285 - 216 = 69 years
Average age of 3 new students = 69 ÷ 3 = 23 years

∴ The average age of the three new students is 23 years.

Question: The sum of the ages of a mother and daughter is 50 years. Five years ago, the mother was four times as old as her daughter. What is the present age of the daughter?

Solution:
ধরি, মেয়ের বর্তমান বয়স = x বছর
তাহলে, মাতার বর্তমান বয়স = (50 - x) বছর

পাঁচ বছর আগে,
মেয়ের বয়স ছিল = (x - 5) বছর
মাতার বয়স ছিল = (50 - x) - 5 = (45 - x) বছর

প্রশ্নমতে,
45 - x = 4(x - 5)
⇒ 45 - x = 4x - 20
⇒ 45 + 20 = 4x + x
⇒ 65 = 5x
⇒ x = 65/5
⇒ x = 13

সুতরাং, মেয়ের বর্তমান বয়স হলো 13 বছর।

Let,
Jashim's present age be x
Jashim's age before 5 years = (x - 5)
Jashim's age after 10 years = (x + 10)
We are given that,
Jashim's age after 10 years (x + 10) is 5 times his age 5 years back (x – 5)

Therefore,
(x + 10) = 5 (x – 5)
⇒ x + 10 = 5x – 25
⇒ 4x = 35
⇒ x = 8.75 years.

The present age of the son is P, he was born P years ago.
The age of the man was: (48 - P).

His age when the son was born should be equal to 2/3 of P.
(48 - P) = (2/3)P
5P = 144
⇒ P = 28.8 years.

বর্তমান বয়স , Johny = 6x এবং mukul = 4x
প্রশ্নমতে,
(6x-5)/(4x-5) = 5/3
20x - 25 = 18x - 15
20x - 18x = 25 -15
2x = 10
x = 5

∴ বর্তমানে Jonny এর বয়স = 6x = 6×5 = 30
উত্তরঃ 30 years

Question: The ages of Ayesha and Rina are in the ratio 5 : 4. After 10 years, the ratio of their ages will be 7 : 6. What is the present age difference between them?

Solution:
Let their present ages be 5x and 4x

According to the question,
(5x + 10)/(4x + 10) = 7/6
⇒ 6(5x + 10) = 7(4x + 10)
⇒ 30x + 60 = 28x + 70
⇒ 30x - 28x = 70 - 60
⇒ 2x = 10
⇒ x = 5

Ayesha’s age = 5x = 25 years
Rina’s age = 4x = 20 years

Difference = 25 - 20 = 5 years

Question: Rafi is currently 18 years old, and his uncle is 54 years old. In how many years will Rafi’s age be one-third of his uncle’s age 6 years from that point?

Solution: 
Given,
Rafi’s current age = 18 years.
Uncle’s current age = 54 years.

Let, the required number of years be x

So, after x years,
Rafi's age = 18 + x
Uncle's age = 54 + x 

Uncle's age 6 years from that point = 54 + x + 6
= 60 + x

According to the question,
18 + x = 1/3(60 + x)
⇒ 3(18 + x) = 60 + x
⇒ 54 + 3x = 60 + x
⇒ 2x = 6
∴ x =3

So, in 3 years, Rafi’s age will be one-third of his uncle’s age 6 years after that.

Question: X is four years older than Y, who is three times as old as Z. If the sum of their ages is 39, then how old is Y?

Solution:
Let the age of Z be = x years
Then, the age of Y = 3x years 
and age of X = (3x + 4) years

According to the question,
(3x + 4) + 3x + x = 39
⇒ 7x + 4 = 39
⇒ 7x = 39 - 4
⇒ 7x = 35
⇒ x = 5

Hence, age of Y = 3x = (3 × 5) years = 15 years

Let A's age be 5x years.
Then, B's age = 3x years.
So,
(5x - 4)/3x + 4 = 1/1
⇒ 5x - 4 = 3x + 4
2x = 8
⇒ x = 4
∴ A's age 4 years hence/B's age 4 years ago = 5x + 4/3x - 4
= (5 × 4 + 4)/(3 × 4 - 4)
= 24/8
= 3/1
= 3:1
Answer: 3:1

Age of Raju when he got the job = 17 + 3 + 2 + 1 = 23 yr

Age of Raju at the time of marriage = 23 + 3 = 26 yr

Let K be Kalam's age and D be his daughter's age
Five years ago, K-5 = 2(D-5)
K - 2D = -5 ....(1)
and their ages are in the ratio 11:6.
i.e., K/D = 11/6
6K = 11D
⇒ K = 11D/6 ....(2)
Sub. K = 11D/6 in (2)
⇒ (11D/6) - 2D = -5
⇒ 11D - 12D = -30
⇒ -D = -30
⇒ D=30
Thus, the age of his daughter is 30 years.
Now, K = 11 x 30/6 = 55 years
Hence the age of Kalam is 55 years.

Question: The average age of a group of 20 employees is 26 years. When 4 more employees join the group, the average age increases by 4 years. What is the average age of the new employees?

Solution:
Average age of 20 employees = 26 years
∴ Total age of 20 employees = 20 × 26 = 520 years

After 4 more employees join:
Total number of employees = 20 + 4 = 24 employees
New average age = 26 + 4 = 30 years
Total age of 24 employees = 24 × 30 = 720 years

∴ Total age of 4 new employees = 720 - 520 = 200 years

∴ Average age of 4 new employees = 200 ÷ 4 = 50 years

Question: The average weight of 12 students is 52 kg. If a new student joins, the average becomes 53 kg. What is the weight of the new student?

Solution:
Average weight of 12 students = 52 
Total weight = 12 × 52
= 624 kg.

Let the weight of the new student = x kg.
New total number of students = 13

New average = 53 
Total weight = 13 × 53
= 689 kg.

Weight of new student = new total - original total
= 689 - 624
= 65 kg

∴ The weight of the new student is 65 kg.

Let daughter’s age be = A
and mother’s age be = B

ATQ, 2A + B = 70 ..... (i)
and, A + 2B = 95 ..... (ii)
(ii)×2 - (i) ⇒ 3B = 120
⇒ B = 40

Given that, six years ago, ratio of the ages of Ashraf and Rahat = 6:5
Hence we can assume that,
age of Ashraf six years ago = 6x
age of Rahat six years ago = 5x
After 4 years, the ratio of their ages = 11:10
⇒ (6x + 10):(5x + 10) = 11:10
⇒ 10(6x + 10) = 11(5x + 10)
⇒ 5x = 10
⇒ x = 10/5
⇒ x = 2
Rahat's present age
=(5x + 6) = 5 × 2 + 6
= 16.

Question: The ages of five people are as follows 36, 48, 52, 40, and 44 years. What should be the age of a sixth person so that the average age of all six becomes 45 years?

Solution:
Let the age of the sixth person be x.
The average of six persons = Total age of 6 persons ÷ 6.

ATQ, 
(36 + 48 + 52 + 40 + 44 + x)/6 = 45
⇒ (220 + x)/6 = 45
⇒ 220 + x = 270
⇒ x = 270 - 220
⇒ x = 50

∴ The age of the sixth person should be 50 years.

Let the age of Robin be y
Roman is 15 years older than Robin = (y + 15).
So Roman's age 5 years ago = (y + 15 – 5)
Robin's age before 5 years = (y – 5)

5 years ago, Roman is 3 times as old as Robin
(y + 15 – 5) = 3 (y – 5)
(y + 10) = (3y – 15)
2y = 25
y = 12.5
Robin's age = 12.5 years
Roman's age = (y + 15)
= (12.5 + 15)
= 27.5 years.

Question: The present age of a father is four times the age of his daughter. After 5 years, the ratio of their ages will be 3 : 1. What is the present age of the father? 

Solution:
Let the daughter’s present age be x years.
Then father’s present age = 4x years.

After 5 years,
∴ (4x + 5)/(x + 5) = 3/1
⇒ 1(4x + 5) = 3(x + 5)
⇒ 4x + 5 = 3x + 15
⇒ 4x - 3x = 15 - 5
⇒ x = 10

∴ Present age of the father = 4x years
= 4 × 10
= 40 years

The present age of Tomal = 5
Present age of Razib = 5 - 2 = 3
Let present age of Akib = x
Then, present age of Razib = (x - 6)/18

According to the question,
(x - 6)/18 = 3
x - 6 = 54
x = 54 + 6
x = 60 years.

Let the present age of the person be x years.

According to the question,
Then, 3(x + 3) - 3(x - 3) = x
⇒ 3x + 9 - 3x + 9 = x
⇒ 18 = x
Hence the present age of the person was 18 years.

Question: X’s age is 20% more than Y’s age. If the sum of their ages if 20 more than the difference of their ages, find their respective ages.

Solution: 
Let X's age be x and Y's age be y

First condition,
X’s age is 20% more than Y’s age.
 ⇒ xx = y + 0.20y
 ⇒ x = 1.20y
 ∴ x = (6/5)y.......(1)

Second condition,
The sum of their ages is 20 more than the difference of their ages.
x + y = (x - y) + 20
⇒ x + y = x - y + 20
⇒ 2y = 20
⇒ y = 20/2
∴ y = 10 years

Now that we have Y's age, substitute  y = 10 into Equation 1
⇒ x = (6/5)y = (6/5) × 10 = 12
∴ x = 12 years

X's age is 12 years and Y's age is 10 years.

Sum of ages of husband, wife and child= (24 × 3) + 9 = 81 year
Sum of ages of wife and child 
=> 25 × 2 + 10 = 50 + 10 = 60 year
Age of husband = 81 - 60 = 21 year

Question: The ratio of a man's age to his son's age is 5 : 2, and the product of their ages is 490. What will the son's age be after 5 years?

Solution: 
Let the man's age = 5x years
And, son's age = 2x years

∴ 5x × 2x = 490
⇒ 10x² = 490
⇒ x² = 49
⇒ x = 7

Son's age = 2 × 7 = 14 years
∴ Son's age after 5 years = 14 + 5 = 19 years

Question: A father is three times as old as his son. After 12 years, he will be twice as old as his son. Find the father’s present age.

Solution:
Let the son’s present age be x years.
Then, the father’s present age = 3x years.

After 12 years,
Son’s age = x + 12
Father’s age = 3x + 12

According to the question,
3x + 12 = 2(x + 12)
⇒ 3x + 12 = 2x + 24
⇒ 3x - 2x = 24 - 12
⇒ x = 12

∴ Father’s present age = 3 × 12 = 36 years.

Let x be Sumon's age, then Akib's age will be 2x
After 5 years their ages will be (x + 5) and (2x + 5) respectively.
Given that the ratio(after 5 Years) is = 6:11
According to the question,
6/11 = (x + 5)/(2x + 5)
⇒ 6(2x + 5) = 11(x + 5)
⇒ 12x + 30 = 11x + 55
⇒ x = 55 - 30
= 25
And 2x = 50
Hence the ages of Sumon and Akib will be 25 and 50.

Let the Father be x years when he died.
Average Age 10 years ago be A.
Total Age 10 years ago = 6 × A
Total Age after 10 years(Just before father's Death) = 6A + 6 × 10 = 6A + 60
Father Died and Baby was born => the Total number of people in the family is Same (6)
Baby born today so age of baby = 0
(6A + 60 - x)/6 = 6A/6
=> A + 10 -(x/6) = A
=> x/6 = 10
=> x = 60
Therefore we can conclude that the father was 60 years old when he died.

Question: A man was asked to state his age in years. His reply was, "Take my age 4 years hence, multiply it by 4 and then subtract 4 times my age 4 years ago, and you will know how old I am". What is the age of the man?

Solution:
Let the age of the man be = p years
Then,
⇒ 4(p + 4) - 4(p - 4) = p
⇒ 4p + 16 - 4p + 16 = p
⇒ p = 32
∴ The age of the man is 32 years.