Problems on Ages

Let,
Fahim's age = x
Bilash's age = y and
Wasim's age = z.

According to the given information
Age of Bilash - Age of Fahim = Age of Fahim - Age of Wasim
⇒ y - x = x - z
⇒ 2x = y + z ......(i)
Also, y + z = 66
from (i) x = 33 years

but, y + z = 66 means, there can be many sets of values that will satisfy the equation.
01 - 65
02 - 64
... ... ...
32 - 34

We want to determine x - z but as we can't get a fixed value for z; hence, the answer can not be determined.

Question: The present ages of three cousins are in the ratio of 4 : 5 : 6. Five years ago, their total age was 60 years. In three years, what will be the age of the youngest cousin?

Solution: Present age ratio of three cousins is 4 : 5 : 6
Let, their age is 4X, 5X and 6X respectively.

ATQ,
4X - 5 + 5X - 5 + 6X - 5 = 60
Or, 15X - 15 = 60
Or, 15X = 75
Or, X = 5

∴ The present age of the youngest cousin is = 4X = 4 × 5 = 20 years.
In three years, his age will be = (20 + 3) = 23 years.

Question: In a group of five men, no two men have the same age. The oldest man is 50 years old, and the youngest is 30 years old. If X is the average age of the men in the group, which of the following best indicates all and only possible values of X? (All ages are in whole numbers)

Solution: 
Here, 
Oldest man = 50 years
Youngest man = 30 years
Average age = X

Since no two people are the same age, the minimum average = (30 + 31 + 32 + 33 + 50)/5
= 176/5
= 35.2 
and the maximum average = (30 + 47 + 48 + 49 + 50)/5
= 224/5
= 44.8

That is, the value of X is greater than 35 and less than 45.
i.e. 35 < X < 45 

Question: A mother was 27 years old when her daughter was born. Currently, the mother's age is 9 years more than three times her daughter's age. How old will the daughter be in 6 years?

Solution:
Let the daughter's current age be x years.
∴ Mother's current age = 3x + 9 years.

According to the question,
3x + 9 - x = 27
⇒ 2x = 27 - 9
⇒ 2x = 18
∴ x  = 9

∴ Daughter's current age 9 years.
∴ Daughter's age after 6 years = 9 + 6 = 15 years.

ATQ,
6x - 5 : 4x - 5 = 5 : 3
Or, 20x - 25 = 18x - 15
Or, 2x = 10
Or, x = 5
So, John's age now is = 6×5 = 30 years

Question: A is five times as old as B. 5 years ago, A was ten times as old as B. What will be the sum of their ages after 8 years?

সমাধান:
ধরি,
B-এর বর্তমান বয়স = x বছর
A-এর বর্তমান বয়স = 5x বছর

5 বছর আগে,
B-এর বয়স ছিল = (x - 5) বছর
A-এর বয়স ছিল = (5x - 5) বছর

প্রশ্নমতে,
5x - 5 = 10(x - 5)
⇒ 5x - 5 = 10x - 50
⇒ 50 - 5 = 10x - 5x
⇒ 45 = 5x
⇒ x = 45/5
∴ x = 9

সুতরাং, B-এর বর্তমান বয়স 9 বছর এবং A-এর বর্তমান বয়স 5 × 9 = 45 বছর।

8 বছর পর,
B-এর বয়স হবে = 9 + 8 = 17 বছর
A-এর বয়স হবে = 45 + 8 = 53 বছর

∴ 8 বছর পর তাদের বয়সের সমষ্টি হবে = 17 + 53 = 70 বছর।

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;
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.

Total Age = 28 + 24 = 52
Average Age = 52/2
= 26

Let,
After T years their age will be the same as before.
ATQ,
24 + T + 28 + T + T + T = 26 × 4 = 104
Or, 4T = 52
Or, T = 52/4
Or, T = 13 years

So after 13 years their age will be the same average as before.

Let the mother's present age be x year.
Then, the person's present age = 2/5 x year.
(2x/5 + 8) = 1/2 (x + 8)
2(2x + 40) = 5(x + 8) 
=> x = 40.

Question: Mr. Babul is aged three times than his son Mahi. After 6 years, he would be two and a half times of Mahi's age. After further 6 years, how many times would he be of Mahi's age?

Solution:
Let Mahi's present age be x years.
Then, Mr. Babul's present age = 3x years 

∴ After 6 years, Mahi's age will be (x + 6) years 
And, after 6 years, Mr. Babul's age will be (3x + 6) years

ATQ,
3x + 6 = (5/2) × (x + 6)
⇒ 3x + 6 = (5x + 30)/2  
⇒ 6x + 12 = 5x + 30
⇒ 6x - 5x = 30 - 12
⇒ x = 18

∴ After 6 years, Mahi's age will be (18 + 6) years
= 24 years
∴ After 6 years, Mr. Babul's age will be (3 × 18) + 6) years
= 54 + 6
= 60 years

∴ After further, 6 years, Mahi's age will be (24 + 6) years
= 30 years
∴ After 6 years, Mr. Babul's age will be (60 + 6) years
= 66 years

∴ Required Ratio = 66/30 
= 11/5 

∴ After further 6 years, Mr. Babul's age will be 11/5 times of Mahi's age.

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

Solution:
Given that,
Today Ratul 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 × Ratul's age

Now,
Ratul'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, Ratul's father will be twice as old as Ratul.

Question: The present ages of Hasan and Kamal are in the ratio 4 : 5. After 6 years, the ratio of their ages will be 5 : 6. What is the difference in their present ages?

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

After 6 years,
Hasan's age = 4x + 6
Kamal's age = 5x + 6

According to the question,
(4x + 6)/(5x + 6) = 5/6
⇒ 6(4x + 6) = 5(5x + 6)
⇒ 24x + 36 = 25x + 30
⇒ 25x − 24x = 36 − 30
⇒ x = 6

Hasan's present age = 4 × 6 = 24 years
Kamal's present age = 5 × 6 = 30 years

∴ Difference = 30 − 24 = 6 years

Question: Currently, the age of P is twice the age of Q. After 6 years, P's age will be 3/2 times the age of Q. What is the present age of P?

Solution:
Let the present age of Q = x years
Then the present age of P = 2x years

After 6 years, age of P = 2x + 6
And age of Q = x + 6

According to the question,
After 6 years, P's age will be 3/2 times Q's age. So,
⇒ 2x + 6 = (3/2) × (x + 6)
⇒ 2(2x + 6) = 3(x + 6)
⇒ 4x + 12 = 3x + 18
⇒ 4x - 3x = 18 - 12
∴ x = 6
Therefore, Present age of Q = 6 years
And present age of P = 2x = 12 years.

So the present age of P is 12 years.

Question: The difference of ages between two brothers is 6 years. If the ratio of their ages is 7 : 5, find their ages

Solution:
Let
The ages of the two brothers be 7x and 5x. (elder : younger).
Difference of ages = 6 

Accordingly, 
7x - 5x = 6
⇒ 2x = 6  
⇒ x = 3.

∴ Elder = 7x = 7 × 3 = 21 
∴ Younger = 5x = 5 × 3 = 15

So Elder brother 21 years and younger brother 15 years.

Question: A is twice as old as B. 12 years ago, A was five times as old as B. Find the present age of A.

Solution:
Let the present age of B is = x years.
Then the present age of A is = 2x years (since A is twice as old as B).

12 years ago,
Age of A = 2x - 12
and age of B = x - 12

According to the problem,
2x - 12 = 5(x - 12)
⇒ 2x - 12 = 5x - 60
⇒ 5x - 2x = - 12 + 60 
⇒ 3x = 48
⇒ x = 48/3
∴ x = 16

∴ Present age of B = 16 years
∴ Present age of A = 2 × 16 = 32 years

16 years ago,
Let, Tania (T) = x years and Grandfather (G) = 8x years.
After 8 years from now,
T = (x + 16 + 8) years and G = (8x + 16 + 8) years.

8x + 24 = 3(x + 24)
⇒ 5x = 48
8 years ago,
T/G = (x + 8)/(8x + 8)
= {(48/5) + 8}/{8 × (48/5) + 8}
= (88/5)/(424/5)
= 88/424
= 11/53

Let the ages of children be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.
Then, x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50
⇒ 5x = 20
⇒ x = 4.
∴ Age of the youngest child = x = 4 years.

Given, Captain's age 26 years and Wicketkeeper's age 29 years.
Let, Average age be X years

ATQ,
9(x - 1) + 26 + 29 = 11x
⇒ 9x - 9 + 26 + 29 = 11x
⇒ 2x = 46
⇒ x = 23

Question: The present ages of three cousins are in the ratio of 5 : 6 : 7. three years ago, their total age was 45 years. In two years, what will be the age of the youngest cousin?

Solution:
Present age ratio of three cousins is 5 : 6 : 7
Let their ages be 5x, 6x, and 7x, respectively

ATQ,
5x - 3 + 6x - 3 + 7x - 3 = 45
⇒ 18x - 9 = 45
⇒ 18x = 54
⇒ x = 3

∴ The present age of the youngest cousin is = 5x = 3 × 5 = 15 years.
In two years, his age will be = (15 + 2) = 17 years

Question: The ratio of the present ages of a father and his son is 5 : 1. Five years ago, the ratio was 9 : 1. What will be the father's age five years from now?
 
Solution:
Let the present age of the son = x years
Then present age of the father = 5x years

Five years ago, Father's age = 5x - 5
And son's age = x - 5

According to the question,
(5x - 5) : (x - 5) = 9 : 1
⇒ (5x - 5)/(x - 5) = 9/1
⇒ 5x - 5 = 9(x - 5)
⇒ 5x - 5 = 9x - 45
⇒ 9x - 5x = 45 - 5
⇒ 4x = 40
⇒ x = 40/4 = 10
∴ x = 10

Therefore, present age of son = 10 years
present age of father = 5 × 10 = 50 years

Five years from now father's age = 50 + 5 = 55 years

So the father's age five years from now will be 55 years.

Let the age of the youngest child is 'X' year.
Then the child elder than the yongest is 'X + 3' years
Since each have 3years difference upto 5 children
And given that
=> X + X+3 + X+6 + X+9 + X+12 = 50
=> 5X + 30 = 50
=> X = 4 years.

Question: The father explained to his son, “When you were born, I had reached the age you are now.” If the father is 52, what was the son’s age five years ago?

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x years

2x = 52
⇒ x = 52/2
= 26 years

∴ the son's age five years back was = 26 - 5
= 21 years

Question: Five years ago, Rahul was twice as old as his sister. Now, Rahul is 6 years older than his sister. What is the present age of Rahul?

Solution:
Let,
Present age of Rahul's sister = x years
Present age of Rahul = x + 6 years (since Rahul is 6 years older)

Five years ago,
Rahul’s age = (x + 6) - 5 = x + 1
And sister’s age = x - 5

According to the question,
x + 1 = 2(x - 5)
⇒ x + 1 = 2x - 10
⇒ 2x - x = 10 + 1
∴ x = 11

∴ Rahul's present age = x + 6 = 11 + 6 = 17 years.