Question

.The age of the father is twice the age of the elder son. Twelve years hence, the
age of the father will be thrice that of the younger son. If the difference

between the elder and younger son is 15, the age of the father is:

A. 21 B. 42 C. 63 D. 84​

Answer 1

Answer :

Option B) 42 years

Explanation :

Let the age of elder son be x years and the age of younger son be y years

Given the age of father is twice the age of elder son i.e.

$\sf \star \: Father's \: age = 2x$

Twelve years hence :

$\bullet \sf \: Age \: of \: father = 2x + 12$

$\bullet \sf \: Age \: of \: younger \: son = y + 12$

Given Twelve years hence, the age of father will be thrice that of the younger son i.e.

$\star \sf \: Father's \: age = 2x + 12 = 3(y + 12)....(i)$

Also, given the difference between the elder and younger son is 15 years i.e.

$\star \sf \: x - y = 15....(ii)$

$\longrightarrow \sf \: x = 15 + y$

Substitute x = 15+y in equation i)

$\longrightarrow \sf \: 2x + 12 = 3(y + 12)$

$\longrightarrow \sf \: 2(15 + y)+ 12 = 3y + 36$

$\longrightarrow \sf \:30 + 2y + 12 = 3y + 36$

$\longrightarrow \sf \:42 + 2y = 3y + 36$

$\longrightarrow \sf \:2y - 3y = 36 - 42$

$\longrightarrow \sf \: - y = - 6$

$\longrightarrow \sf \boxed{ \red{ \sf \: y = 6}}$

Putting the value of y = 6 in equation ii)

$\star \sf \: x - y = 15$

$\longrightarrow \sf \: x - 6 = 15$

$\longrightarrow \sf \:x = 15 + 6$

$\longrightarrow \sf \boxed{ \red{ \sf \: x = 21}}$

Hence,

• Younger son's age = 6 years

• Elder son's age = 21 years

• Father's age = 2(21) = 42 years

Answer 2

Given :

• The age of the father is twice the age of the elder son.
• Twelve years hence, the
• age of the father will be thrice that of the younger son.
• The difference between the elder and younger son is 15

To Find :

• Present age of Father.

Solution :

Let the present age of father be x years.

Let the present age of elder son be y years.

Let the present age of younger son be z years.

Case 1 :

The father's age is twice the elder son's age.

Equation :

$\sf{x=2y\:\:\:(1)}$

Case 2 :

The age of Father after 12 years will be thrice that of younger son.

Age of father after 12 years, (x+12) years.

Age of younger son after 12 years, (z+12) years.

Equation :

$\longrightarrow$ $\sf{x+12=3(z+12)}$

$\longrightarrow$ $\sf{x+12=3z+36}$

$\longrightarrow$ $\sf{x-3z=36-12}$

$\longrightarrow$ $\sf{x-3z=24}$

From equation (1),

$\longrightarrow$ $\sf{2y-3z=24\:\:\:(2)}$

Case 3 :

The difference between the age of elder son and younger son is 15.

Equation :

$\longrightarrow$ $\sf{y-z=15}$

$\longrightarrow$ $\sf{y=15+z}$

Substitute this value of y in equation (2),

$\longrightarrow$ $\sf{2(15+z)-3z=24}$

$\longrightarrow$ $\sf{30+2z-3z=24}$

$\longrightarrow$ $\sf{30-z=24}$

$\longrightarrow$ $\sf{-z=24-30}$

$\longrightarrow$ $\sf{-z=-6}$

$\longrightarrow$ $\sf{z=6}$

Substitute, z = 6 in equation (2),

$\longrightarrow$ $\sf{2y-3(6)=24}$

$\longrightarrow$ $\sf{2y-18=24}$

$\longrightarrow$ $\sf{2y=24+18}$

$\longrightarrow$ $\sf{2y=42}$

$\longrightarrow$ $\sf{y=\dfrac{42}{2}}$

$\longrightarrow$ $\sf{y=21}$

Substitute, y = 21 in equation (1),

$\longrightarrow$ $\sf{x=2y}$

$\longrightarrow$ $\sf{x=2(21)}$

$\longrightarrow$ $\sf{x=42}$

$\large{\boxed{\sf{Present\:age\:of\:father\:=\:42\:years}}}$