## Problem Based on ages

#### Problem Based on ages

1. Five years ago, Mayank's mother was three times as old as Mayank. After five years she will be twice as old as Mayank. The present age of Mayank is :

1. Method 1
Let us assume the present age of Mayank is X and his mother age is Y.
Five years ago, Mayank's age = X - 5
Five years ago, Mayank's mother age = Y - 5

After five years, Mayank's age = X + 5
After five years , Mayank's mother age = Y + 5

Method 2
Let us assume the present age of Mayank be P, then the 5 years before Mayank's age = P - 5
According to Question
Age of his mother before 5 years = 3 x (P - 5 )
Present age of his mother = 3 x (P - 5 ) + 5
After 5 Years Mayank's age = P + 5
After 5 Years Mother' age = 3 x (P - 5 ) + 5 + 5

##### Correct Option: B

Method (1)
Let us assume the present age of Mayank is X and his mother age is Y.
Five years ago, Mayank's age = X - 5
Five years ago, Mayank's mother age = Y - 5
According to question,
Five years ago, Mayank's mother was three times as old as Mayank.
Y - 5 = 3 (X - 5)
⇒ Y - 5 = 3X - 15
⇒ 3X - Y = 15 - 5
⇒ 3X - Y = 10................(1)
Again According to question,
After five years, Mayank's age = X + 5
After five years , Mayank's mother age = Y + 5
After five years she will be twice as old as Mayank.
Y + 5 = 2 ( X + 5 )
⇒ Y + 5 = 2X + 10
⇒ Y - 2X = 10 - 5
⇒ Y - 2X = 5 ....................(2)
Add the equation (1) and (2), we will get
3X - Y + Y - 2X = 10 + 5
X = 15
Put the vale of X in equation (2), We will get
Y - 2 x 15 = 5
Y + 30 + 5 = 35 years
Mayank' Present Age is 15 years and mother age is 35 year.

Method 2
Let us assume the present age of Mayank be P, then the 5 years before Mayank's age = P - 5
According to Question
Age of his mother before 5 years = 3 x (P - 5 )
Present age of his mother = 3 x (P - 5 ) + 5
After 5 Years Mayank's age = P + 5
After 5 Years Mother' age = 3 x (P - 5 ) + 5 + 5
Now again according to question,
After five years she will be twice as old as Mayank, then
3 x (P - 5 ) + 5 + 5 = 2 (P + 5)
⇒ 3P - 5 = 2P + 10
⇒ 3P - 2P = 10 + 5
⇒ P = 15
Mayank's present Age is 15 years.

1. The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was four times the father's age at that time. What are their present ages?

1. Method 1
Lt us assume the present age of father is P and son age is Q.
According to question,
P + Q = 45

Method 2
Let the present ages of father and his son be x and (45 - x) years respectively.

##### Correct Option: B

Lt us assume the present age of father is P and son age is Q.
According to question,
P + Q = 45.................. (1)
Five years ago, the father age = P - 5
Five years ago, the son age = Q - 5
According to question,
Five years ago, the product of their ages was four times the father's age at that time.
(P - 5) (Q - 5) = 4 (P - 5)
PQ - 5P - 5Q + 25 = 4P - 20
PQ - 5P - 5Q + 25 - 4P + 20 = 0
⇒ PQ - 9P - 5Q + 45 = 0
⇒ P(Q - 9) - 5(Q - 9) = 0
⇒ (Q - 9) (P - 5) = 0
⇒ (Q - 9) = 0 or (P - 5) = 0
⇒ Q = 9 or P = 5
Put the vale of Q in equation (1), we will get.
if Q = 9 then P = 45 - 9 = 36 which is matched our answer.

If P = 5 then Q = 45 - 5 = 40 years which is not match our answer.

Method 2
Let the present ages of father and his son be x and (45 - x) years respectively;
then according to question,
(x - 5) (45 - x - 5) = 4 (45 - x -5)
⇒ x - 5 = 4
⇒ x = 9
Now, Father's age = 45 - x = 45 - 9 = 36 years

1. The difference between the present ages of Arun and Barun is 14 years. 7 years ago the ratio of their ages was 5 : 7 respectively. The present age of Barun is :

1. Let the present ages of Arun and Barun are x and x + 14 years respectively.

##### Correct Option: D

Let the present ages of Arun and Barun are x and x +14 years respectively; then
According to question, 7 years ago the ratio of their ages was 5 : 7
(x - 7)/(x + 14 - 7) = 5/7
⇒ (x - 7)7 = 5(x+14 - 7)
⇒ 7x - 49 = 5x + 35
⇒ 2x = 84
∴ x = 42
Hence, age of Barun = 42 + 14 = 56 years

1. The ratio between the ages of Ram and Mohan is 4 : 5 and that between Mohan and Anil is 5 : 6. If sum of the ages of three be 90 years, what is the age of Mohan ?

1. Here, ratio of ages of Ram, Mohan and Anil = 4 : 5: 6
Let their present ages be 4x, 5x and 6x years respectively.

##### Correct Option: D

Here, ratio of ages of Ram, Mohan and Anil = 4 : 5: 6
Let their present ages be 4x, 5x and 6x years respectively.
then 4x + 5x + 6x = 90
⇒ 15x = 90
∴ x = 6
Hence, age of Mohan = 5 x 6 = 30 years

1. Ten years ago, the age of Divya was half of the age of Namrata. If the ratio of present ages of both is 3 : 4, the sum of their present ages is:

1. Let the present age of Divya and Namrata are 3x and 4x years respectively;

##### Correct Option: A

Let the present age of Divya and Namrata are 3x and 4x years respectively; then
According to question,
Ten years ago, Age of Divya = 3x - 10
Ten years ago, Age of Namrata = 4x - 10
Now According to question,
Ten years ago, the age of Divya was half of the age of Namrata
3x - 10 = (4x - 10)/2
⇒ 6x - 20 = 4x - 10
⇒ 2x =10
∴ x = 5
Hence, sum of their ages = 3x + 4x = 7x =7 x 5 = 35 years