Problems on Ages

Question: The present age of Mr. Salman is three times the age of his son. Six years hence , the ratio of their ages will be 5 : 2. What is the present age of Mr. Salman?

Solution:
Let the son's age be x years ,
Then Mr. Salman's age = 3x years
Then, 
∴ (3x + 6)/(x + 6) = 5/2
⇒2(3x + 6) = 5(x + 6)
⇒ 6x + 12 = 5x + 30
⇒ 6x - 5x = 30 - 12
⇒ x = 18

∴ Present age of Mr. Salman = 3x years
= (3 × 18) years
= 54 years

Question: Fuad, Rubel and Pavel are cousins. Fuad’s age is one-third of Rubel and Pavel is five years elder than Rubel. If the sum of the age of the cousins is 40, find the ages of Fuad.

Solution:
Let,
Rubel’s age = x
Fuad’s age = x/3
Pavel’s age = x + 5

ATQ,
Fuad + Rubel + Pavel = 40
(x/3) + x + (x + 5) = 40
⇒ (x/3) + 2x + 5 = 40
⇒ (x + 6x)/3 = 40 - 5
⇒ 7x/3 = 35
⇒ x = (35 × 3)/7 = 15
∴ x = 15 years

∴ Fuad age is = 15 × (1/3) = 5 years.

Let daughter’s age be x and father’s age be 3x.
Father's age is 3 times more aged than his daughter,
therefore father’s present age = x + 3x = 4x
After 5 years,
The father’s age is 3 times more than his daughter's age.
(4x + 5) = 3 (x + 5)
(4x+5) = 3 (x+5)
(4x + 5) = 3 (x + 5)
x = 10

After 5 years it was (4x + 5),
then after further 5 years, father’s age = (4x +10) and daughter’s age = (x + 10)
(4x + 10)/(x + 10)
= [{(4 × 10) + 10}/(10 + 10)]
= 50/20
= 5/2
= 2.5
After further 5 years, the father will be 2.5 times of daughter’s age.

According to Rakib 17th, 18th or 19th ...... (i)
According to her sister 19th, 20th, 21st or 22nd ......(ii)
From (i) and (ii) ⇒ 19th
Answer: 19

Question: If 5 years ago, the ratio of age of Hasan and Rina was 1 : 2 and after 15 years from present their ratio would be 5 : 6. Find the age of Rina after 20 years.

Solution:
Let, present age of Hasan be x
and present age of Rina be y.

Then, according to question
(x - 5)/(y - 5) = 1/2
⇒ 2x - 10 = y - 5
⇒ x = (y + 5)/2 .............(1)

Also,
(x + 15)/(y + 15) = 5/6
⇒ 6x + 90 = 5y + 75
⇒ 6x + 15 = 5y

Putting value of x from equation 1, we get
3y + 15 + 15 = 5y
⇒ 2y = 30
⇒ y = 15

∴ Age of Rina after 20 years = 15 + 20 = 35 years.

Let Benu’s age now be B.
Anika’s age now is A.
(A - 6) = P(B - 6)
But A is 17 and therefore 11 = P(B - 6)
11/P = B - 6
(11/P) + 6 = B

Question: B’s age after 12 years would be equal to 4 times his age 6 years ago. Find his age 5 years hence?
 
Solution:
Let B’s present age be ‘x’ years.

According to the question,
x + 12 = 4(x - 6)
⇒ x + 12 = 4x - 24
⇒ 3x = 36
⇒ x = 12

∴ B’s present age = 12 years

Therefore, B’s age 5 years hence = 12 + 5 = 17 years

Question: The difference between the ages of a father and his son is 40% of the father’s age. If the son is 18 years old, find the father’s age.

Solution:
Let
Let father’s age = x

Then,
x - 18 = 40% of x
⇒ x - 18 = 40x/100
⇒ x - 18 = 2x/5
⇒ x - 2x/5 = 18
⇒ (5x - 2x)/5 = 18
⇒ 3x = 90
∴ x = 30

So the father’s age is 30 years.

Let the present age of A be a years and that of B be b years
Then, 4 years ago,
A's age = (a - 4)
B's age = (b - 4)
Now, according to the given information in question,
{(a - b)/2}/4(b - 4) = 5/12 or (a - 4)/2(4b - 16) = 5/12 or (a - 4)/(4b - 16) = 5/6
By cross multiplying we get
or, 6a - 24 = 20b - 80
or, 6a - 20b = -56
or, 10b - 3a = 28
After 8 years, 
(a + 8)2 + 2 = b + 8
or, a/2 + 4 + 2 = b + 8
or, b - a/2 = -2
or, 2b - a = -4 .......(i)
a = 2b + 4 ......(ii)
Putting the value of a in equation (i), we get
10b - 3(2b + 4) = 28
or, 4b = 40
∴ b = 10
Hence, the present age of B is 10 years.

Let Rajeev's present age be  x year.
Rajeev's age after 15 year = (x + 15) year.
Rajeev's age 5 year back = (x - 5) year
Then ATQ, 
x + 15 = 5 (x - 5)
x + 15 = 5x - 25
=> x = 10
Hence, Rajeev's present age = x = 10 year.

Let the age of son = X years

∴Age of mother would be =2X

As per question 20 years ago;
10 (X -20) = 2X - 20
⇒ 10X - 200 = 2X - 20
⇒ 10X - 2X= - 20 + 200
⇒ 8X = 180
⇒ X= 180/8
= 22.5 years

∴Age of mother = 22.5 × 2 = 45 years.

Question: A is four years older than B, who is thrice as old as C. If the total of the ages of A, B, and C is 46, how old is A?

Solution:
Let
C's age be = a years
Then, B's age = 3a years
A's age = (3a + 4) years

∴ (3a + 4) + 3a + a = 32
⇒ 7a + 4 = 46
⇒ 7a = 42
⇒ a = 6

Hence, A's age = (3 × 6) + 4 = 22 years.

Let,
7 years ago son’s age was x years old and father’s age was 5x year.
Now,
(x+7)/(5x+7) = 1/3,
so x = 7
Son’s age = 7 + 7 = 14
Father’s age= 5 × 7 + 7 = 42
After 7 years,
Son’s age : Father’s age = 21:49 = 3:7

Question: The present ages of three persons are in the ratio of 4 : 7 : 9. Six years ago, the sum of their ages was 62. What was the age of the oldest person six years ago? 

Solution: 
Let their present ages are 4x, 7x and 9x years respectively. 
∴ 6 years ago, their present ages are (4x - 6), (7x - 6) and (9x - 6) respectively. 

ATQ,
(4x - 6) + (7x - 6) + (9x - 6) = 62 
⇒ 20x - 18 = 62
⇒ 20x = 62 + 18
⇒ x = 80/20
⇒ x = 4

∴ Their present ages are (4 × 4) =16, (7 × 4) = 28  and (9 × 4) = 36 years respectively.
∴ Age of the oldest person six years ago is (36 - 6) = 30 years. 

Question: A mother told her daughter, "I was your present age at the time of your birth." If the mother is currently 38 years old, what was the daughter’s age five years ago? 
Solution: 
Let,
Daughter present age = x years
Mother's present age = (x + x) years
= 2x years

ATQ,
2x = 38
⇒ x = 19
So daughter age five years back was = 19 - 5
= 14 years.
 

Question: A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

Solution: 
Let,
'x' years ago, the ratio of their ages was 3 : 2

ATQ,
(50 - x) : (35 - x) = 3 : 2
⇒ (50 - x)/(35 - x) = 3/2
⇒ 105 - 3x = 100 - 2x
⇒ 3x - 2x = 105 - 100
∴ x = 5

Let Roni's present age be x years. Then, father's present age =(x + 3x) years = 4x years.
there4 (4x + 8) = (5/2)(x + 8)
=> 8x + 16 = 5x + 40
=> 3x = 24
=> x = 8.

Hence, required ratio = (4x + 16)/(x + 16)
= 48/24
= 2

মনেকরি,
5 বছর পূর্বে পিতার বয়স = 5x বছর
এবং পুত্রের বয়স =x বছর
∴ বর্তমানে পিতার বয়স = (5x+5) বছর
এবং পুত্রের বয়স = (x + 5) বছর
প্রশ্নমতে, 5x + 5 + 2 = 3 (x + 5 + 2)
⇒ 5x + 7 = 3(x + 7)
⇒ 5x + 7 = 3x + 21
⇒ 2x = 14
⇒ x=7
∴ বর্তমানে পিতার বয়স 5×7+5 = 40 বছর
বর্তমানে পুত্রের বয়স = 7 + 5 = 12 বছর
∴ পিতা ও পুত্র = 40 : 12 =10 : 3

Question: Today is Hasan's 12^th birthday and his father's 40^th birthday. How many years from today will Hasan's father be twice as old as Hasan at that time?

Solution:
Given that,
Today, Hasan is 12 years old
And his father is 40 years old

Age difference = 40 - 12 = 28 years (constant)

Let after x years, father's age = 2 × Hasan's age

Now,
Hasan's age after x years = 12 + x
Father's age after x years = 40 + x

ATQ,
40 + x = 2 × (12 + x)
⇒ 40 + x = 24 + 2x
⇒ x = 40 - 24

∴ x = 16

So, in 16 years, Hasan's father will be twice as old as Hasan.

Question: The present age of a father is 5 times that of his son. After 8 years, the ratio will be 3 : 1. Find their age gap.

Solution:
Let the present age of the son = x years.
Present age of the father = 5x years.

After 8 years:
Son = x + 8 
Father = 5x + 8
Given ratio after 8 years = 3 : 1

Accordingly,
(5x + 8​) / (x + 8) = 3
⇒ 5x + 8 = 3(x + 8)
⇒ 5x + 8 = 3x + 24
⇒ 5x - 3x = 24 - 8
⇒ 2x = 16
∴ x = 8

Present ages of
Son = x = 8 years
And Father = 5x = 40 years

Age gap = 40 - 8 = 32

So the age gap is 32 years.

Let the son's age 18 years ago be x years,
Then man's age 18 years ago = 3x years
(3x+18) = 2(x+18)
⇒ 3x+18 = 2x+36
⇒ x = 18
Sum of their present ages
⇒ (3x+18+x+18) years
⇒ (4x+36) years
⇒ (4×18+36) years
⇒ 108 years

The age of the daughter = (3 × 28) - (2 × 38)
= 84 - 76
= 8 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.