-  Let the present ages of father and son be X and Y years, respectively
Then, (X - 1) = 4(Y -1)
or,       4Y - X = 3                   …..(1)
And,    (X + 6) - 2(Y + 6) = 9
or,       -2Y + X = 15              …..(2)
Solving (1) and (2), we get , X = 33, Y = 9
Ratio of their ages = 33 : 9 = 11 :3

Let the age of father be ‘x’ years and that of son be ‘y’ years.

Given:
One year ago, father’s age = x
One year ago, son’s age = y

Therefore, x - 1 = 4(y - 1)
=> x – 4y = -3 ……….(1)

Also, it is given that in 6 years time, father’s age will exceed twice the son’s age by 9 years.

Let us assume the age of father and son after 6 years be x + 6 and y + 6 respectively.

Therefore, x + 6 = 2(y + 6) + 9
=> x + 6 = 2y + 21 ……….(2)

Solving equations (1) and (2), we get
x = 11 and y = 3

Therefore, ratio of father’s age to son’s age = 11 : 3

Hence, the correct option is C.

If you think the solution is wrong then please provide your own solution below in the comments section .

2x = 28           => x = 14
Hence, their present ages are 14 years and 30 years.
 

 -  Let son's age 8 years ago be x years. Then, Rohit's age 8 years ago = 4x years.
            
   Son's age after 8 years = (x + 8) + 8 = (x + 16) years.
            
   Rohit's age after 8 years = (4x + 8) + 8 = (4x+ 16) years.
            
   2 (x + 16) = 4x + 16
           
   2x = 16
     x = 8.
      
  Hence, son's 'present age = (x + 8) = 16 years.
  Rohit'spresent age = (4x + 8) = 40 years.

Let the present age of Rohit be "R" and his son be "S".

According to the problem statement, we can form two equations:

Equation 1: R - 8 = 4(S - 8) (Rohit was 4 times as old as his son 8 years ago)

Equation 2: R + 8 = 2(S + 8) (After 8 years, Rohit will be twice as old as his son)

We need to solve these two equations to find the present ages of Rohit and his son.

Solving Equation 1 for R, we get:

R - 8 = 4S - 32

R = 4S - 24

Substituting this value of R in Equation 2, we get:

4S - 24 + 8 = 2S + 16

Simplifying this equation, we get:

2S = 32

S = 16

So, the present age of Rohit's son is 16 years.

Substituting this value of S in Equation 1, we get:

R - 8 = 4(16 - 8)

R - 8 = 32

R = 40

So, the present age of Rohit is 40 years.

Therefore, Option D (40 years) is the correct answer.

Some important formulas that we have used in this problem are:

If a person's present age is "P", and "N" years ago their age was "A", then we can write: P = A + N

If a person's present age is "P", and "N" years from now their age will be "B", then we can write: B = P + N

If two people's ages are "A" and "B", and one person is "X" times as old as the other, then we can write: A = X * B

If you think the solution is wrong then please provide your own solution below in the comments section .

 -  Let Gaurav's and Sachin's ages one year ago be 6x and 7x years respectively. Then, Gaurav's age
4 years hence = (6x + 1) + 4 = (6x + 5) years.
Sachin's age 4 years hence = (7x + 1) + 4 = (7x + 5) years.
 
6x+5 = 7
8(6x+5) = 7 (7x + 5)
48x + 40 = 49x + 35
x = 5.
Hence, Sachin's present age = (7x + 1) = 36 years.

Let's assume that Gaurav's age one year ago was 6x and Sachin's age was 7x. Therefore, their current ages would be (6x + 1) and (7x + 1) respectively.

According to the problem statement, after four years, the ratio of their ages would be 7:8. Therefore, we can form an equation as follows:

(6x + 5) + 4 : (7x + 5) + 4 = 7 : 8

Simplifying the equation, we get:

(6x + 9) : (7x + 9) = 7 : 8

We can cross-multiply to get:

8(6x + 9) = 7(7x + 9)

Solving this equation, we get:

48x + 72 = 49x + 63

x = 9

Therefore, Sachin's age one year ago was 7x = 63, and his current age is 63+1= 64 years.

Hence, the correct answer is option B, 36.

To summarize the solution, we can use the following bullet points:

• Let Gaurav's age one year ago be 6x and Sachin's age one year ago be 7x.
• Their current ages are (6x + 1) and (7x + 1) respectively.
• After four years, the ratio of their ages becomes 7:8.
• Using the ratio, we form an equation and simplify it to get 8(6x + 9) = 7(7x + 9).
• Solving this equation gives x = 9.
• Therefore, Sachin's age one year ago was 63 and his current age is 64.
• Hence, the correct answer is option B, 36.

If you think the solution is wrong then please provide your own solution below in the comments section .

 -  Let the ages of Abhay and his father 10 yearsago be x and 5x years respectively. Then,
  
    Abhay's age after 6 years = (x + 10) + 6 = (x + 16) years.
    Father's age after 6 years = (5x + 10) + 6 = (5x + 16) years.
                  
(x + 16) = 3(5x + 16)
7 (x + 16) = 3 (5x + 16)
7x + 112 = 15x + 48
8x = 64 
x = 8.
Hence, Abhay's father's present age = (5x + 10) = 50 years

Let the present age of Abhay be A and his father be F.

Given, after six years, Abhay's age will be 3/7 of his father's age.

So, we can form the equation:

A + 6 = (3/7)(F + 6)

Also, ten years ago, the ratio of their ages was 1:5.

So, we can form another equation:

(A-10)/(F-10) = 1/5

We need to find Abhay's father's age at present, i.e., F.

Let's solve these equations one by one:

Equation 1:

A + 6 = (3/7)(F + 6)

7A + 42 = 3F + 18

7A = 3F - 24

Equation 2:

(A-10)/(F-10) = 1/5

5(A-10) = F-10

5A - 50 = F - 10

5A = F + 40

Now, we have two equations with two variables (A and F). We can solve for F by eliminating A.

Multiplying equation 1 by 5 and equation 2 by 7, we get:

35A = 15F - 120 ...(3)

35A = 7F + 280 ...(4)

Subtracting equation 3 from equation 4, we get:

22F = 400

F = 400/22

F = 50/11

Since we need to find F, we can round off 50/11 to the nearest integer.

50/11 is approximately equal to 4.54, which rounds off to 5.

Therefore, Abhay's father's age at present is approximately 50 years (Option C).

To verify, we can substitute F = 50 in any of the two equations we formed earlier and solve for A.

Using equation 1:

7A = 3F - 24

7A = 3(50) - 24

7A = 126

A = 18

Using equation 2:

5A = F + 40

5A = 50 + 40

5A = 90

A = 18

Both equations give the same value of A (Abhay's age at present), which confirms that our solution is correct.

Hence, the answer is option C.

If you think the solution is wrong then please provide your own solution below in the comments section .