Problems on Ages

Total age of 8 children = 8 x 12 = 96 yr
Total age of 7 children = 12 + 8 + 14 + 11 + 9 + 13 + 15 = 82
The age of 8th child = 96 - 82 = 14 yr

Let the present age be x years.
20 years ago my age was x - 20.

According to given data,
x - 20 = x/3
=> 2x = 60
=> x = 30

A's age = 44 × (6/11) years = 24 years and
B's age = (44 - 24) years = 20 years.
Ratio of their ages after 8 years = (24 + 8)/(20 + 8)
= 32/28
= 8/7
= 8:7

Bambi’s present age = 6x,
Samba’s present age = 4x
Therefore,
(6x - 5)/(4x - 5) = 5/3
Or, 20x - 25 = 18x - 15
So, x = 5
And samba’s age = 6 × 5 = 30

Question: The sum of the present ages of a son and his father is 70 years. After 5 years, the age of the father will be three times that of the son. At present, their ages are?

Solution:
Let,
The son's age is x years.
Then, father's age = (70 - x) years.

ATQ,
(70 - x) + 5 = 3(x + 5)
⇒ 75 - x = 3x + 15
⇒ 4x = 60
∴ x = 15

∴ Son's age = 15 years
And the father's age = (70 - 15) = 55 years.

Question: P is five years younger than Q, who is twice as old as R. If the sum of their ages is 55, how old is Q?

Solution:
Let,
the age of R be = x years
Then,
the age of Q = 2x years
and age of P = (2x – 5) years

According to the question,
(2x – 5) + 2x + x = 55
⇒ 5x – 5 = 55
⇒ 5x = 60
⇒ x = 12

Hence, age of Q = 2x = (2 × 12) years = 24 years

Let the age of sushma be x and 
the age of her son is y
Then five year before x-5=5(y-5) ...(1)
Five year hence x+5 = 3(y+5)-8 .....(2)
By soving (1) & (2), we get
5y - 15 = 3y + 7
y = 11 
=> x = 35
Therefore, the age of Sushma = 35 and her son = 11.

Question: A son's present age is three-sevenths of his father’s age. After 7 years, he will be half of his father’s age. Find the present age of the father.

Solution:
Let, the father's present age be x years

Then, the son's present age (3x/7) years

∴ (3x/7) + 7 = (1/2) × (x + 7)
⇒ (3x + 49)/7 = (1/2) × (x + 7)
⇒ 7x + 49 = 6x + 98
⇒ x = 49

∴ the present age of the father is 49 years.

Sum of the present ages of husband, wife and child = {(27 × 3) + (3 × 3)} years = 90 years.

Sum of the present ages of wife and child = {(20 × 2) + (5 × 2)} years = 50 years.

Husband's present age = (90 - 50) years = 40 years

Question: A man's age is now 3 times his son's age. Eight years back, the man's age was 5 times his son's age. What is the present age of the son?

Solution:
Let Man's son's present age = x
Man's present age = 3x

According to the question
5(x - 8) = 3x - 8
⇒ 5x - 40 = 3x - 8
⇒ 5x - 3x = 40 - 8
⇒ 2x = 32
x = 16 

A + B + C = 30×3 = 90 yrs ...... (1)
ATQ,
B - A = C - B
Or, A+C = 2B
From (1),
B + 2B = 90
Or, B = 30
∴ D = B + 40 = 30 + 40 = 70
So, B + D = 30+70 = 100 yrs

Question: The sum of the ages of a father and son is 68 years. Four years ago, the father was five times as old as his son. What is the present age of the son?

Solution:
ধরি, পুত্রের বর্তমান বয়স = x বছর
তাহলে, পিতার বর্তমান বয়স = (68 - x) বছর

চার বছর আগে,
পুত্রের বয়স ছিল = (x - 4) বছর
পিতার বয়স ছিল = (68 - x) - 4 = (64 - x) বছর

প্রশ্নমতে,
64 - x = 5(x - 4)
⇒ 64 - x = 5x - 20
⇒ 64 + 20 = 5x + x
⇒ 84 = 6x
⇒ x = 84/6
⇒ x = 14

সুতরাং, পুত্রের বর্তমান বয়স হলো 14 বছর।

Question: The average age of P, Q, R, S and T is 32 years. The average age of P and Q is 28 years, and the average age of R and S is 35 years. What is the age of T?

Solution:
Given that, 
There are 5 people P, Q, R, S, T
Average age of all five = 32 years
Total age of all five = 5 × 32 = 160 years

And, 
Average age of P and Q = 28 years
So, age of (P + Q) = 2 × 28 = 56 years

Average age of R and S = 35 years
So, age of (R + S) = 2 × 35 = 70 years

∴ Now, total age of (P + Q + R + S) = 56 + 70 = 126 years

Age of T = Total age of all five - Age of (P + Q + R + S)
= 160 - 126
= 34 years

So the age of T is 34 years.

Question: A brother was 7 years senior to his sister 9 years ago. Their ages together add up to 53 at present. Find the sister’s and brother's current age.

Solution: 
Let the brother's present age be b and the sister's present age be s.

9 years ago:
Brother's age: b - 9
Sister's age: s - 9

Given,
 b - 9 = (s - 9) + 7
⇒ b - 9 = s - 2
⇒ b = s + 7

b + s = 53
⇒ b = s + 7
⇒ (s + 7) + s = 53
⇒ 2s + 7 = 53
⇒ 2s = 46
⇒ s = 23

The sister's present age is 23 years.

Brother's present age is 30 years.

According to the question,
A + B = 60,
A = 2B
2B + B = 60
⇒ B = 20 then A = 40.

5 years,
their ages will be 45 and 25.
Sum of their ages = 45 + 25
= 70.

Let present ages of A, B, and C be 4x, 7x, and 9x respectively.
(4x − 8) + (7x − 8) + (9x 8) = 56
20x = 80
x = 4

Hence present ages of A, B, and C are (4 × 4), (7 × 4), and (9 × 4)
That is 16, 28, and 36.

Question: The age of the father ten years ago was thrice the age of his son. Ten years hence, the father's age will be twice the age of his son. What is the ratio of their present ages?

Solution:
Let,
son's age 10 years ago be x years.
Then, father’s age 10 years ago = 3x years.

Son's present age = (x + 10) years
Father's present age = (3x + 10) years.

according to the question,
(3x + 10) + 10 = 2 (x + 10 + 10)
⇒ 3x + 20 = 2 (x + 20) 
⇒ 3x + 20 = 2x + 40 
∴ x = 20

Ratio of present ages of father and the son
= (3x + 10)/(x + 10)
= {(3 × 20) + 10}/(20 + 10)
= 70/30 
= 7 : 3

If ages in the numerical are mentioned in the ratio A : B, then A: B will be Ax and Bx
1) At present : Ratio of their ages = 5 : 3. Therefore, 5 : 3 will be 5x and 3x.
Rohan's age 4 years ago = 5x – 4
Rahul's age after 4 years = 3x + 4

2)Ratio of Rohan’s age 4 years ago and Rahul's age after 4 years is 1: 1
Therefore,
(5x - 4)/(3x + 4) = 1/1
⇒ 5x - 4 = 3x + 4
⇒ 5x - 3x = 4 + 4
⇒ 2x = 8
⇒ x = 4
Solving, we get x = 4

3)We are asked to find the ratio between Rohan's age 4 years hence and Rahul's age 4 years ago.
Rohan's age : (5x + 4)
Rahul's age: (3x – 4)
Ratio of Rohan's age and Rahul’s age
(5x + 4)/(3x - 4) = 24/8
(5x + 4)/(3x - 4) = 3/1
(5x + 4) : (3x - 4) = 3 : 1.

Let A’s age 10 years ago be = x years
Then, B’s age 10 years ago = 2x years
ATQ, (x + 10) / (2x + 10) = 3/4
Or, 6x + 30 = 4x + 40
Or, x = 5
So, the total of their present ages = (x + 10 + 2x + 10) = (5 + 10 + 2.5 + 10) = 35 years.

Laxmi present age = 51 - 8 = 43
So Madhu Age = (43 - 7) = 36
So, Diviya Present Age = 36 × 7/6 = 42
So Difference = (42 - 36) = 6 years

Sum of their ages = 40 × 5 = 200 years
Sum of A & B’s ages = 35 × 2 = 70 years
Sum of C & D’s ages = 42 × 2 = 84
So, Age of E = 200 - (70 + 84) = 46

Question: Rifat is twice as old as his friend Sifat. Sifat is 5 years older than Rafi. In 5 years, Rifat will be three times as old as Rafi. How old is Sifat now?

Solution:
Let
Rifat’s age = R
Sifat’s age = S
Rafi’s age = F

Now,
Rifat is twice as old as Sifat
∴ R = 2S
Sifat is 5 years older than Rafi
S = F + 5
∴  F = S - 5

In 5 years, Rifat will be three times as old as Rafi.
That means after 5 years,
R + 5 = 3(F + 5)
⇒ 2S + 5 = 3(S - 5 + 5)
⇒ 2S + 5 = 3S
∴ S = 5

So Sifat is 5 years old now.

Question: The sum of the ages of a mother and her daughter is 50 years. The mother’s age is four times the daughter’s age. After how many years will the mother’s age be three times the daughter’s age?

Solution:
Let the daughter's current age be x.
Then, the mother's current age is 4x.

ATC,
x + 4x = 50
⇒ 5x = 50
∴ x = 10

∴ Daughter's current age x = 10 years
and Mother's current age is 4x = 4 × 10 = 40 years

Let, the number of years after which mother’s age will be three times the daughter’s age be t.

So after t years,
The daughter’s age will be 10 + t
The mother’s age will be 40 + t

ATC,
40 + t = 3(10 + t)
⇒ 40 + t = 30 + 3t
⇒ 40 - 30 = 3t - t
⇒ 10 = 2t
⇒ t = 5

∴ After 5 years, the mother’s age will be three times the daughter’s age.

Question: The sum of the ages of A, B, and C is 120 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of A?

Solution:
Let, A, B, and C's ages 10 years ago be 2x, 3x, and 4x years respectively

ATQ,
(2x + 10) + (3x + 10) + (4x + 10) = 120
⇒ 9x  + 30 = 120
⇒ 9x = 90
⇒ x = 10

So, the age of A = 2 × 10 + 10 = 30

Question: A father is 30 years older than his son. In 10 years, the father will be twice the son’s age. Find the present age of father.

Solution:
Let the present age of the son be x years.
Then the present age of the father = x + 30 years.

In 10 years,
Son’s age = x + 10
Father’s age = x + 30 + 10 = x + 40

According to the problem, father’s age will be twice the son’s age,
x + 40 = 2(x + 10)
⇒ x + 40 = 2x + 20
⇒ 2x - x = 40 - 20
⇒ x = 20

∴ Present ages of father = x + 30 = 20 + 30 = 50

Sum of 18 observations = 18 x 124
=2232

Correct sum of 18 observations = 2232 - 46 - 82 + 64 + 28
= 2196
Therefore,
Correct average =2196/18
= 122.

Mobin's age:Xam's age = 4:5 = 1:5/4
Mobin's age:Faisal's age = 5:6 = 1:6/5
Let, Mobin's age be x years. Then, Xami's age = 5x/4
And, Faisal's age = 6x/5 years
∴ x + 5x/4 + 6x/5 = 69
⇒ 20x + 25x + 24x = 69 × 20
⇒ 20x + 25x + 24x = 1380
⇒ 69x = 1380
⇒ x = 1380/69
= 20
Xami's age = 5x/4 = (5 × 20)/4
= 25 years

Question: One year ago, the ratio between Fahim and Rohan's ages was 3 : 2. One year hence, the ratio of their age will be 4 : 3. What is the sum of their present ages in years?

Solution:
ধরি, এক বছর আগে ফাহিমের বয়স ছিল 3x বছর এবং রোহানের বয়স ছিল 2x বছর।
বর্তমানে ফাহিমের বয়স = (3x + 1) বছর
বর্তমানে রোহানের বয়স = (2x + 1) বছর

এক বছর পর তাদের বয়সের অনুপাত হবে 4 : 3
∴ এক বছর পর ফাহিমের বয়স হবে = (3x + 1) + 1 = (3x + 2) বছর
∴ এক বছর পর রোহানের বয়স হবে = (2x + 1) + 1 = (2x + 2) বছর

প্রশ্নমতে,
(3x + 2)/(2x + 2) = 4/3
⇒ 3(3x + 2) = 4(2x + 2)
⇒ 9x + 6 = 8x + 8
⇒ 9x - 8x = 8 - 6
⇒ x = 2

সুতরাং, বর্তমানে ফাহিমের বয়স = (3 × 2) + 1 = 6 + 1 = 7 বছর
এবং বর্তমানে রোহানের বয়স = (2 × 2) + 1 = 4 + 1 = 5 বছর
∴ তাদের বর্তমান বয়সের যোগফল = 7 + 5 = 12 বছর।

Question: The ages of five persons are 54, 82, 45, 55, and 59 years. What should be the age of a sixth person so that the average age becomes 61 years?

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

Accordingly:
(54 + 82 + 45 + 55 + 59 + x) / 6​ = 61
⇒ (295 + x) / 6 = 61
⇒ 295 + x = 366
⇒ x = 366 - 295
⇒ x = 71

Let's father's age x and son's age is y
ATQ, x + y = 42 ....(i)
x - y = 22 .... (ii)
(i) - (ii) = 2y = 20
∴ y = 10
∴ son's age is 10 years

Question: Eight years ago, Karim was five times as old as his son. Now, Karim is 28 years older than his son. What is the present age of the son?

Solution:
ধরি,
বর্তমানে Karim-এর বয়স = x বছর এবং তার ছেলের বয়স = y বছর

৮ বছর আগে, Karim তার ছেলের বয়সের পাঁচ গুণ ছিল। অর্থাৎ
⇒ x - 8 = 5(y - 8).......... (১)

আবার,
বর্তমানে Karim তার ছেলের চেয়ে ২৮ বছর বড়।
x = y + 28

x এর মান (১) এ বসিয়ে পাই,
⇒ (y + 28) - 8 = 5(y - 8)
⇒ y + 20 = 5y - 40
⇒ 20 + 40 = 5y - y
⇒ 60 = 4y
∴ y = 15

∴ ছেলের বর্তমান বয়স = 15 বছর।

Question: Today is Karim's 18th birthday and his father's 48th birthday. How many years from today will Karim's father be twice as old as Karim at that time?

Solution:
Given that, Today Karim is 18 years old
And his Father is 48 years old
Age difference = 48 - 18 = 30 years (constant)

Let After x years, father's age = 2 × Karim's age

Now, Karim's age after x years = 18 + x
Father's age after x years = 48 + x

ATQ,
48 + x = 2(18 + x)
⇒ 48 + x = 36 + 2x
⇒ 2x - x = 48 - 36
⇒ x = 12

So, in 12 years, Karim's father will be twice as old as Karim.

Question: Find the mother age after 8 years, if the ratio of age of mother and son is 5 ∶ 2 and the product of their ages in years is 1000.

Solution:
Given that,
Mother : Son = 5 ∶ 2
Product of ages = 1000

Let ages of mother and son be 5x, 2x

According to problem,
5x × 2x = 1000
⇒ 10x² = 1000
⇒ x² = 100 = 10²
∴ x = 10

So, age of mother after 8 years = (5x + 8) 
= (5 × 10 + 8)
= 50 + 8
= 58 years

∴ The mother age is 58 years.

Question: Trisha’s grandfather was 8 times older to her 16 years ago. He would be 3 times her age 8 years from now. Eight years ago, what was the ratio of Trisha’s age to that of her grandfather? 

Solution: 
Let, trisha was x years old 16 years ago.
 
grandfather was 8x years 

ATQ, 
8x + 16 + 8 = 3 (x + 16 + 8)
⇒ 8x + 24 = 3x + 72 
⇒ 8x - 3x = 72 - 24 = 48 
⇒ x = 48/5 = 9.6 

 Eight years ago, trisha was 9.6 + 8 = 17.6 
grandfather was = 8 × 9.6 + 8 = 84.8 

ratio = 17.6 : 84.8 
= 176 : 848
= 11 : 53 

Let C's age be x year. 
Then, B's age = 2x year. 
A's age = (2x + 2) year. 
(2x + 2) + 2x + x = 27 
5x = 25 
=> x = 5 
Hence, B's age = 2x = 10 year.

Let the age of the son before 8 years ago = x
Then, age of Kamal before 8 years age = 4x
After 8 years, Kamal will be twice as old as his son
(4x + 16) = 2(x + 16)
⇒ 4x - 2x = 32 - 16
⇒ 2x = 16
⇒ x = 8 years.
The present age of kamal = 4x + 8
= (4 × 8) + 8
= 32 + 8
= 40 years.

Question: The present ages of Kamal and Jamal are in the ratio 5 : 7. After 8 years, the ratio of their ages will be 3 : 4. What is the difference in their present ages?

Solution:
Let their present ages be 5x and 7x.

After 8 years, Kamal's age = 5x + 8
Jamal's age = 7x + 8

According to the question,
(5x + 8)/(7x + 8) = 3/4
⇒ 4(5x + 8) = 3(7x + 8)
⇒ 20x + 32 = 21x + 24
⇒ 20x - 21x = 24 - 32
⇒ - x = - 8
∴ x = 8

∴ Kamal's present age = 5 × 8 = 40 years
∴ Jamal's present age = 7 × 8 = 56 years

∴ Difference = 56 - 40 = 16 years

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

Mother'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, 8 years ago,
the age of A and B was 4x and 5x respectively. Age of C = y
so, the present age of A and B is 4x + 8 and 5x + 8 respectively 

A/Q,
5x + 8 : y = 4 : 5 ------------ (1)
y - (4x + 8) = 20 ----------- (2)
Solving this equation we will get x = 8 and y = 60

So, present age of Age of A = 4×8 + 8
= 40
and present age of B = 5×8 + 8 
= 48
From equation (1),
C = {(5x + 8) × 5} ÷ 4 = 60

So, the total age of A, B & C= 40 + 48 + 60
= 148

Question: The sum of the ages of a man and his son is 70 years. After 10 years, the ratio of their ages will be 2 : 1. What is the son's present age?

Solution:
ধরি,
বাবার বর্তমান বয়স x বছর
ছেলের বর্তমান বয়স y বছর

প্রশ্নমতে,
x + y = 70 ............ (1)

আবার, 10 বছর পর, 
(x + 10)/(y + 10) = 2/1
⇒ x + 10 = 2(y + 10)
⇒ x + 10 = 2y + 20
⇒ x - 2y = 10 ............. (2)

Equation, (1) - (2)
x + y - x + 2y = 70 - 10
⇒ 3y = 60
⇒ y = 20

∴ পুত্রের বর্তমান বয়স = 20 বছর

Question: Age of Kamal will be 4 times the age of Rina in 6 years from today. If ages of Kamal and Mahi are 7 times and 6 times the age of Rina respectively, what is present age of Kamal?

Solution:
Let,
Rina's present age = R years
Then,
Kamal's present age = 7 × R = 7R
Mahi's present age = 6 × R = 6R

Age of Kamal will be 4 times the age of Rina in 6 years.
Kamal’s age after 6 years = 7R +6
Rina’s age after 6 years = R + 6

ATQ,
7R + 6 = 4(R + 6)
⇒ 7R + 6 = 4R + 24
⇒ 7R - 4R = 18
⇒ R = 18/3
∴ R = 6

∴ Kamal's present age = 7 × R = 7R = 7 × 6 = 42 years

Question: Rina is currently four times older than Meena. Six years ago, Rina was ten times older than Meena. What will be the sum of their ages after 5 years?

Solution:
ধরি,
বর্তমানে মিনার বয়স = x বছর
∴ রিনার বয়স = 4x বছর

৬ বছর পূর্বে,
মিনার বয়স (x - 6) বছর.
রিনার বয়স (4x - 6) বছর.

ATC,
4x - 6 = 10(x - 6)
⇒ 4x - 6 = 10x - 60
⇒ 6x = 54
∴ x = 9

∴ মিনার বর্তমান বয়স = 9 বছর
রিনার বর্তমান বয়স = 4 × 9 = 36 বছর

5 বছর পরে,
মিনার বয়স = 9 + 5 = 14 বছর
রিনার বয়স = 36 + 5 = 41 বছর

∴5 বছর পরে উভয়ের বয়সের যোগফল = 14 + 41 = 55 বছর