Problems on Ages

Question: The average age of Arif, Tania, Joya, and Rafi is 22 years. The average age of Arif, Tania, and Joya is 20 years and the average age of Tania, Joya, and Rafi is 24 years. Find the average age of Tania and Joya.

Solution:
Given,
Arif + Tania + Joya + Rafi = 22 × 4 = 88 years ……(1)
Arif + Tania + Joya = 20 × 3 = 60 years ……(2)
Tania + Joya + Rafi = 24 × 3 = 72 years ……(3)

From (1) - (2)⇒
Rafi’s age = 88 − 60 = 28 years

From (3)⇒
Tania + Joya = 72 - 28 = 44 years

∴ Average age of Tania and Joya = 44/2 = 22 years

Question: Aisha is five times as old as Rohan. 8 years ago, Aisha was thirteen times as old as Rohan. What will be the sum of their ages after 4 years?

Solution:
Let,
The present age of Rohan is x years.
∴ The present age of Aisha is 5x years.

ATC,
5x - 8 = 13(x - 8)
⇒ 5x - 8 = 13x - 104
⇒ - 8x = - 96
⇒ x = 12

So, Rohan's current age = 12 years.
Aisha's current age = 5 × 12 = 60 years.

Rohan's age after 4 years = 12 + 4 = 16 years
Aisha's age after 4 years = 60 + 4 = 64 years

∴ Sum of their ages After 4 years = 16 + 64 = 80 years.

We are given that,
age of father 10 years ago was 3 times the age of her son
So,
let the age of the son be x and as the father's age is 3 times the age of her son, let it be 3x, three years ago.
At present,
Father's age will be (3x + 10) and son's age will be (x + 10)
After 10 years,
Father's age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Father's age is twice that of son
(3x + 10) +10 = 2 [(x + 10) + 10]
(3x + 20) = 2[x + 20]
Solving the equation, we get x = 20
We are asked to find the present ratio.
(3x + 10) : (x + 10) = 70 : 30
(3x + 10) : (x + 10) = 7 : 3.

Let N be the no. of persons in the group.
Required number of person is given by;
Member in group × aged increased = difference of replacement
N × 5 = 38 - 18
Or, 5N = 20
Or, N = 4

Question: P's age 3 years ago was three times the present age of Q. At present R's age is twice the age of Q. Also R is 12 years younger than P. What is the present age of R?

Solution:
Let the present age of Q be a years
Three years ago P's age = 3a years
Then, present age of P is (3a + 3)
R's present age = 2a

According to question
(3a + 3) - 2a = 12
a = 9 year

∴ Present age of R = 2a = 2 × 9 = 18 years

প্রশ্ন: মা ও মেয়ের বর্তমান বয়সের অনুপাত ৭ : ২। ৫ বছর পর তাদের বয়সের অনুপাত হবে ৮ : ৩। ১০ বছর পর মেয়ের বয়স কত হবে?

সমাধান:
ধরি,
মায়ের বর্তমান বয়স = ৭ক বছর
মেয়ের বর্তমান বয়স = ২ক বছর

৫ বছর পর,
মায়ের বয়স হবে = ৭ক + ৫
মেয়ের বয়স হবে = ২ক + ৫

প্রশ্নানুসারে,
(৭ক + ৫)/(২ক + ৫) = ৮/৩
⇒ ৩(৭ক + ৫) = ৮(২ক + ৫)
⇒ ২১ক + ১৫ = ১৬ক + ৪০
⇒ ২১ক - ১৬ক = ৪০ - ১৫
⇒ ৫ক = ২৫
∴ ক = ৫

∴ মেয়ের বর্তমান বয়স = ২ক = ২ × ৫ = ১০ বছর
∴ ১০ বছর পর মেয়ের বয়স = ১০ + ১০ = ২০ বছর

Let, a daughter’s present age be x year.
In that case mother present age would be = 30x years
30x + 18 = 3 (x + 18)
⇒ 27x = 36
⇒ x = 4/3
∴ So the present age of mother = (30 × 4/3)
= 40 years.

Let current ages of X and Y correspondingly, is 6A & 5A
Given: 6A + 5A = 44
=> A = 4
Proportion of ages after 0.8 decades will be
6A + 8 : 5A + 8
32:28 (or) 8:7

Question: Six years ago, Hena was 1/4^th of the age she will be after 12 years from now. How old is she now?

Solution:
ধরি
হেনার বর্তমান বয়স = x বছর
6 বছর পূর্বে হেনার বয়স ছিলো = x - 6 বছর
12 বছর পর হেনার বয়স হবে = x + 12 বছর

প্রশ্নমতে
x - 6 = (1/4)(x + 12)
বা, 4x - 24 =  x + 12 
বা, 4x - x = 24 + 12
বা, 3x = 36
∴ x = 12
হেনার বর্তমান বয়স = 12 বছর

Let, Samir’s age is = x
So Babul’s age is = x - 2

ATQ,
x - 2 = 2(x - 2 - 2)
Or, x - 2 = 2x - 8
∴ x = 6

Question: Ratul is four times as old as Tarek. 12 years ago, Ratul was 10 times as old as Tarek. What will be the sum of their ages after 5 years?

Solution:
ধরি, বর্তমানে Tarek-এর বয়স = x বছর।
তাহলে Ratul-এর বয়স = 4x বছর।

12 বছর আগে তাদের বয়স ছিল:
Tarek = x - 12 বছর
Ratul = 4x - 12 বছর

প্রশ্নমতে,
4x - 12 = 10(x - 12)
⇒ 4x - 12 = 10x - 120
⇒ 120 - 12 = 10x - 4x
⇒ 108 = 6x
⇒ x = 108/6
∴ x = 18

∴ বর্তমানে Tarek এর বয়স = 18 বছর,
Ratul এর বয়স = 4 × 18 = 72 বছর

5 বছর পর তাদের বয়স হবে:
Tarek = 18 + 5 = 23 বছর
Ratul = 72 + 5 = 77 বছর

∴ 5 বছর পর তাদের বয়সের যোগফল = 23 + 77 = 100 বছর

ধরি,
ববির বর্তমান বয়স B বছর
আরিফের বর্তমান বয়স A বছর
সামির বর্তমান বয়স S বছর

প্রশ্নমতে, 
A=4B ............(১) 
A+4=36 ..................(২) 
B+6=2(S+6) ...............(৩) 

(২) নং সমীকরণ থেকে পাই, 
A = 36−4 = 32

এখন, A = 4B প্রতিস্থাপন করলে,
32 = 4B
∴ B = 8

(৩) নং সমীকরণ থেকে পাই, 
8+6 = 2(S+6)
14 = 2S+12
2S = 2
S = 1

∴ ৬ বছর পর সামির বয়স = S + 6
= 1 + 6
= 7 বছর। 

3 year ago the age was = 27×3 = 81
Currently sum of their age is  = 81 + 9 = 90
5 year ago = 20×2 = 40
Now, the sum is = 40 + 10 = 50

So, Present age of the husband = 90 - 50 = 40

Let, John's present age = 6x,
and, Mary’s present age = 4x
Therefore,
(6x - 5)/(4x - 5) = 5/3
Or, 20x - 25 = 18x - 15
Or,  x = 5
∴ John’s age = 6 × 5 = 30 years

Let father's age be x year and son's age be y year.
According to question,
2(x+y) = 8y _______(I)
and (x+y)/2 = 30
=> x+y = 60 year_______(II)
From equation (I) and (II)
8y = 120
y = 15 year,
Hence x = 45 year.

Question: Currently, a mother’s age is five times that of her son. After 10 years, her age will be three times her son’s age. What is the son’s present age?

Solution:
Let,
The son’s present age = x years
Then, mother’s present age = 5x years

According to the question,
5x + 10 = 3(x + 10)
⇒ 5x + 10 = 3x + 30
⇒ 5x - 3x = 30 - 10
⇒ 2x = 20
⇒ x = 10

∴ The son’s present age is 10 years.

Question: 10 years ago, Person X was 6 years younger than Person Y. The sum of their present ages is 50. Find the present age of X.

Solution:
Let present age of X = a years  
Present age of Y = b years  

Given that,
a + b = 50 .........(1)
And, 
10 years ago, X was 6 years younger than Y  
⇒ (a - 10) = (b - 10) - 6  
 ⇒ a - 10 = b - 16  
 ∴ a = b - 6  

Now put x = y - 6 in equation (1) then we get,
⇒ (b - 6) + y = 50  
⇒ 2b - 6 = 50  
⇒ 2b = 56  
⇒ b = 56/2 = 28  
∴ b = 28

∴ Present age of X = b - 6 = 28 - 6 = 22 years.

Let,
Maala’s age = 4A and
Kala’s age = 3A
Then,
4A + 3A = 28
A = 4
Maala’s age = 16 years
and Kala’s age = 12 years
Proportion of their ages after 8 is = (16 + 8) : (12 + 8)
= 24 : 20
= 6 : 5

Let present age of the elder person = x
The present age of the younger person = x - 16
According to the question,
x - 6 = 3(x - 16 - 6)
⇒ x - 6 = 3x - 66
⇒ 2x = 60
⇒ x = 30
Hence, the present age of the elder person is 30 years.

Let the age of Ali, Gazi and Masud be respectively 2x, 3x and 4x years

ATQ, 10 years ago, 2x + 3x + 4x = 93 - 30
9x = 63
x = 7
So, 10 years ago Masud's age was, 4x = 28 years

Now, Masud's age is = 38 years

Question: Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age?

Solution:
Let Ronit's present age be x years.
Then, father's present age =(x + 3x) years
= 4x years

Now
4x + 8 = (5/2)(x + 8)
8x + 16 = 5x + 40
8x - 5x = 40 - 16
3x = 24
x = 8

Further 8 years will means 16 years from the current age.
After 16 years from present, Ronit's age be (8 + 16) = 24 years
After 16 years from present, father's age be (8 × 4 + 16) = 48 years

After 16 years from present the ratio of their ages will be 
His father's age/ Ronit's age = 48/24 
His father's age/ Ronit's age =2
His father's age = 2 × Ronit age

Total age of the members = 16 × 22 = 352 years
Total age of 15 members at least = 15 × 19 = 285 years
∴ Possible maximum age = 352 - 285 = 67 years

Let x and y be the ten's and unit's digits respectively of the numeral denoting the woman's age.

Then, woman's age = (10X + y) years;
So, husband's age = (10y + x) years.
Therefore (10y + x) - (10X + y) = (1/11) (10y + x + 10x + y)
⇒ (9y - 9x) = (1/11)(11y + 11x) = (x + y)
⇒ 10x = 8y
⇒ x = (4/5)y

Clearly, y should be a single-digit multiple of 5, which is 5.
So, x = 4, y = 5.
Hence, woman's age = 10x + y = 45 years.

Question: The average of the ages of a man and his daughter is 34 years. If the respective ratio of their ages four years from now is 14 : 5. What is daughter's present age ?

Solution:
Average age of man and his daughter = 34 years
Their total age = (34 × 2) years = 68 years
Let man's age be x years. Then, daughter age = (68 - x) years
∴ (x + 4)/(68 - x + 4) = 14/5
⇒ 5(x + 4) = 14(72 - x)
⇒ 5x + 20 = 1008 - 14x
⇒ 19x = 988
⇒ x = 52

∴ Daughter's present age = (68 - 52) = 16 years.

Question: The ratio of the ages of a father and his son is 5 : 2 respectively. Six years ago, the ratio of their ages was 3 : 1 respectively. What is the son's present age?

Solution:
ধরি, পিতার বর্তমান বয়স = 5x বছর
এবং পুত্রের বর্তমান বয়স = 2x বছর

ছয় বছর আগে,
পিতার বয়স = 5x − 6 বছর
পুত্রের বয়স = 2x − 6 বছর

শর্তমতে,
(5x − 6)/(2x − 6) = 3/1
⇒ 6x - 18 = 5x - 6
⇒ 6x - 5x = 18 - 6
⇒ x = 12

∴ পুত্রের বর্তমান বয়স = 2x = 2 × 12 = 24 বছর

Question: A couple has a son and a daughter. The age of the father is three times the daughter's age and the age of the mother is twice the son's age. The father is 9 years older than the mother and the daughter is 6 years younger than the son. What is the age of the mother?

Solution:
Let the ages of father, mother, son, and daughter be F, M, S, and D respectively.

According to the question (ATQ):
F = 3D .............(i)
M = 2S .............(ii)
F = M + 9 ...........(iii)
S = D + 6 ...........(iv)

From (ii) we get,
M = 2S
Or, M = 2(D + 6) [from (iv)]
Or, M = 2{(F/3) + 6} [from (i)]
Or, M = (2/3) × (F + 18)
Or, M = (2/3) × (M + 9 + 18) [from (iii)]
Or, M = (2/3) × (M + 27)
Or, 3M = 2M + 54
Or, M = 54

∴ The age of the mother is 54 years.

16 years ago, let V = x years and G = 8x years.
After 8 years from now, V = (x + 16 + 8)years and G = (8x + 16 + 8) years.
∴ 8x + 24 = 3 (x + 24)
⇒ 8x - 3x = 72 - 24
⇒ 5x = 48
8 years ago, V/g = (x + 8)/(8x + 8)
= {(48/5) + 8}/{8 × (48/5) + 8}
= (48 + 40)/(384 + 40) = 88/424
= 11/53
Answer: none of these.

Let, the present age of Mira is x
so her sons' present age is x/30
18 year later, the age of Mira will x+18 and the age of her son will (x/30)+18

ATQ,
x + 18 = 3{(x/30)+18}
or, x + 18= (3x/30) + 3.18 = (x/10) + 54
or, 10x + 180 = x + 540 [Multiplying both sides by 10]
or, 10x - x = 540 - 180
or, 9x = 360
or, x = 360/9
or, x = 40
So, The present age of Mira is 40.

Total age of grandparents = 67 × 2
Total age of parents = 35 × 2
Total age of grandchildren = 6 × 3

Total age of all persons = 67 × 2 + 35 × 2 + 6 × 3 = 222
Total number of persons = 2 + 2 + 3 = 7

The average age of the family
= 222/7
= 31(5/7)

Question: A father is five times as old as his son. After eight years, the father will be three times as old as his son will be then. What is the father's present age?

Solution:
Let the son's present age be x years.
Then, the father's present age = 5x years.

After 8 years, 
Son’s age = x + 8
Father’s age = 5x + 8

According to the question,
5x + 8 = 3(x + 8)
⇒ 5x + 8 = 3x + 24
⇒ 5x - 3x = 24 - 8
⇒ 2x = 16
∴ x = 8

So, father’s present age = 5 × 8 = 40 years.

Question: The average age of a group is 28. If a 42-year-old person leaves, the average becomes 26. How many people are in the group?

Solution: 
Let
The original number of people be x.

Total age of the group:
= 28 × x
= 28x

After the 42-year-old leaves, total age:
= 28x - 42

Number of people remaining = x - 1
New average = 26

Accordingly, 
(28x - 42) / (x - 1) = 26
⇒ 28x - 42 = 26(x - 1)
⇒ 28x - 42 = 26x - 26
⇒ 28x - 26x = - 26 + 42
⇒ 2x = 16
⇒ x = 8

In the question, the first piece of relevant information given is that eighteen years ago a father was three times as old as his son. Mathematically an equation can be developed from this but some variables needs to be assigned before we can do this.
Let father’s current age = x (in years)
Let son’s current age = y (in years)
x - 18 = 3(y - 18)
⇒ 3y - 54 = x - 18
⇒ 3y = x + 36
∴ y = x/3 + 12 … (1)

The second piece of information is that currently the father is twice as old as his son. Mathematically a second equation can be derived from this and simultaneous equations can be used where there are two variables and two equations.
x = 2y … (2)
put (1) into (2)
⇒ x = 2(x/3 + 12)
⇒ x = 2x/3 + 24
⇒ x/3 = 24
⇒ x = 72
⇒ x = 2y
⇒ y = x/2
⇒ y = 72/2
∴ y = 36
The father’s age is 72 years old and the son’s age is 36 years old.
Therefore, sum of present ages of son and father = 36 + 72 = 108 years.

Let the average of 8 men be x.
Age of new man be N.
Total age of 8 men = 8x
⇒ x + 4 = ((8x – 30) + N)/8
⇒ 8x + 32 = 8x – 30 + N
⇒ N = 32 + 30
⇒ N = 62

Question: A couple has a son and a daughter. The age of the father is four times the son and the age of the mother is three times the daughter. The mother is 6 years younger to the father and the daughter is 3 years older than the son. What is the age of the mother?

Solution:
Let, father, mother, son and daughter have age of F, M, S and D respectively.
ATQ,
F = 4S ………….(i)
M = 3D………….(ii)
M = F - 6………..(iii)
D = S + 3………..(iv)

From (ii) we get,
M = 3D
Or, M = 3(S + 3)  [from (iv)]
Or, M = 3{(F/4) + 3}  [from (i)]
Or, M = ¾(F + 12)
Or, M = ¾(M + 6 + 12)  [from(iii)]
Or, M = ¾(M + 18)
Or, 4M = 3M + 54
Or, M = 54

∴ the age of the mother is 54 years.

From the given data,
when my age is 24, my mothers age is double of my age 
=> 48 years is my mothers age
=> Difference is 48 - 24 = 24 year.
When my age is 44
=> My mother is 44 + 24 = 68 year aged.

Total age of 10 students = 150 years
Total age of 15 students = 240 years
Total age of 5 new students = 240 - 150 = 90 years
Therefore,
Average age of 5 new students = 90/5 = 18 years

Question: B is twice as old as A. C is twice as old as B. If the difference between ages of A and C is 12 years, find the age of B.

Solution:
Let A’s age = x years
Then,
B’s age = 2x
And C’s age = 2 × B = 2 × 2x = 4x

Given condition, Difference between ages of A and C = 12 years
C - A = 12
⇒ 4x - x = 12
⇒ 3x = 12
∴ x = 4

∴ B’s age = 2x = 2 × 4 = 8 years

Question: The present age of a son is one-fourth of the present age of his father. Eight years ago,  the father's age was seven times the age of his son. What is the present age of the son?

Solution: Let the father's age be 4x years
Then, Son's age = x years

ATQ,
4x - 8 = 7(x - 8)
or, 4x - 8 = 7x - 56
or, -8 + 56 = 7x - 4x
or, 48 = 3x
or, x = 16

∴ The present age of the son = 16 years.