Question

2 years ago, the father age was 6 times his son's age. 6 years hence ratio between the ages of
father and the son's age 10:3.What is the father's present age?
A)40
B)42
C)44
D)48​

Answer 1

$\Large{\underline{\underline{\mathfrak{\red{\bf{Solution}}}}}}$

$\Large{\underline{\mathfrak{\orange{\bf{Given}}}}}$

• 2 years ago, the father age was 6 times his son's age
• 6 years hence ratio between the ages of father and the son's age 10:3

$\Large{\underline{\mathfrak{\orange{\bf{Find}}}}}$

• Present age of both.

$\Large{\underline{\underline{\mathfrak{\red{\bf{Explanation}}}}}}$

Let,

• Present Age of Father = x year
• Present Age of his son = y year

Case(1).

(2 years ago, the father age was 6 times his son's age)

➩(x-2) = 6 * (y-2)

➩ x - 6y = -12 + 2

➩ x - 6y = -10 ----------------(1)

Case(2).

(6 years hence ratio between the ages of father and the son's age 10:3)

➩ (x+6) : (y+6) = 10 : 3

➩ (x+6)/(y+6) = 10/3

➩ 3 * (x+6) = 10 * (y+6)

➩ 3x - 10y = 60 - 18

➩ 3x - 10y = 42 ----------------(2)

Multiply by 3 in equ(1)

➩ 3x - 18y = -30 ---------------(3)

Subtract equ(2) & equ(3)

➩ -10y + 18y = 42 + 30

➩ 8y = 72

➩ y = 72/8

➩ y = 9

Keep value of y in equ(2)

➩ 3x - 10 * (9) = 42

➩ 3x = 42 + 90

➩ 3x = 132

➩ x = 132/3

➩ x = 44

$\Large{\underline{\underline{\mathfrak{\red{\bf{Hence}}}}}}$

• Present Age of Father be (x) = 44 years
• Present Age of his Son be (y) = 7 years.

$\Large{\underline{\underline{\mathfrak{\blue{\bf{Since}}}}}}$

• Your answer will be option Number (c).

__________________

Answer 2

Let ,

• The present ages of father and son be " x " and " y "

First Condition :

2 years ago, the father age was 6 times his son's age

(x - 2) = 6(y - 2)

x - 2 = 6y - 12

x - 6y = - 10 --- (i)

Second Condition :

After 6 years , the ratio between the ages of father and the son's age will be 10 : 3

(x + 6)/(y + 6) = 10/3

3x + 18 = 10y + 60

3x - 10y = 42 --- (ii)

Multiply eq (i) by 3 , we get

3x - 18y = -30 --- (iii)

Subtract eq (iii) from eq (ii) , we get

3x -10y - (3x - 18y) = 42 - (-30)

-10y + 18y = 42 + 30

8y = 72

y = 72/8

y = 9

Put the value of y = 9 in eq (i) , we get

x - 6(9) = -10

x - 54 = -10

x = -10 + 54

x = 44

$\sf \therefore \underline{ {The \: present \: ages \: of \: father \: and \: son \: are \: 44 \: years \: and \: 9 \: years }}$