Question

33. Eight years ago Gautam's age was 4 times older than his. After
eight years Gautam will be 2 times older than his son. Find the age
ratio in their present age of both of them.​

Answer 1

Solution :

$\bf{\red{\underline{\underline{\bf{Given\::}}}}}$

Eight years ago, Gautam's age was 4 times older than his son. After eight years Gautam will be 2 times older than his son.

$\bf{\red{\underline{\underline{\bf{To\:find\::}}}}}$

The age ratio in the present age of both of them.

$\bf{\red{\underline{\underline{\bf{Explanation\::}}}}}$

Let the present age of Gautam's age be R years

Let the present age of son's age be M years

A/q

$\bf{\large{\underline{\underline{\mathcal{EIGHT\:YEARS\:AGO\::}}}}}$

The age of Gautam's = (R-8) years.

The age of Son's = (M-8) years.

So;

$\mapsto\tt{R-8=4(M-8)}\\\\\mapsto\tt{R-8=4M-32}\\\\\mapsto\tt{R-4M=-32+8}\\\\\mapsto\bf{R-4M=-24.........................(1)}$

$\bf{\large{\underline{\underline{\mathcal{AFTER\:EIGHT\:YEARS\::}}}}}$

The age of Gautam's = (R+8) years.

The age of Son's = (M+8) years.

So;

$\mapsto\tt{R+8=2(M+8)}\\\\\mapsto\tt{R+8=2M+16}\\\\\mapsto\tt{R-2M=16-8}\\\\\mapsto\tt{R-2M=8}\\\\\mapsto\bf{R=8+2M...................................(2)}$

Putting the value of R in equation (1),we get;

$\mapsto\tt{8+2M-4M=-24}\\\\\mapsto\tt{8-2M=-24}\\\\\mapsto\tt{-2M=-24-8}\\\\\mapsto\tt{-2M=-32}\\\\\mapsto\tt{M=\cancel{\dfrac{-32}{-2} }}\\\\\mapsto\tt{\pink{M=16\:years}}$

Putting the value of M in equation (2),we get;

$\mapsto\tt{R=8+2(16)}\\\\\mapsto\tt{R=8+32}\\\\\mapsto\tt{\pink{R=40\:years}}$

Thus;

The present age of Gautam's R = 40 years.

The present age of Son's M = 16 years.

&

Their ratio is = 40 : 16

Their ratio is = 5 : 2

Answer 2

Question :

Eight years ago Gautam's age was 4 times older than his son.

After eight years Gautam will be 2 times older than his son.

Solution :

From the attachment we can find that X = 8

Gautam's age = 4x + 16 = 64 years.

Son's age = 24 years.