Question

The ages of the younger son, the elder son and their father are in continuous proportion today. If the younger son is 6 years old and the sum of the ages of the elder son and tge father is 72 years, how old is the elder son today?

Answer 1

Answer:

Present age of elder son in 18 years.

Step-by-step explanation:

Given,

Age of father + age of elder son = 72 years

Age of father = 72 years - age of elder son

Let the age of elder son be a years.So, age of father should be ( 72 - a ) years{ that is 72 years - age of elder son, from above }.

Here,

Age of younger son is 6 years.

Ages of the younger son, the elder son and their father are in continuous proportion today.

Since their ages are in continued proportion, ratio of age of younger son to the age of elder son should be equal to the ratio of age of elder son to the age of father.

It means :

$\sf \implies \dfrac{age\:of\:younger\:son}{age\:of\:elder\:son}=\dfrac{age\:of\:elder\:son}{age\:of\:father}$

= > 6 years / a years = a years / ( 72 - a ) years

= > 6 / a = a / ( 72 - a )

= > 6( 72 - a ) = a^2

= > 432 - 6a = a^2

= > a^2 + 6a - 432 = 0

= > a^2 + ( 24 - 18 )a - 432 = 0

= > a^2 + 24a - 18a - 432 = 0

= > a( a + 24 ) - 18( a + 24 ) = 0

= > ( a - 18 )( a + 24 ) = 0

= > a = 18 or - 24

Since a is assumed as age, it can't be negative. Thus, value of a is 18.

Hence the present age of elder son in 18 years.

Answer 2

Answer:

$\large{\underline{\boxed{\sf 18 \: years. }}}$

Step-by-step explanation:

Given that ;

The age of younger son = 6 years.

And, the sum of ages of elder son and father is 72 years.

_______________________

Let the age of elder son be x years. So,

⇒ x + father's age = 72

⇒ Father's age = (72 - x)years.

Also, it is given that ;

The ages of the younger son, the elder son and their father are in continuous proportion today.

That is,

$\sf \dfrac{Age \: of \: Younger\:son}{Age \: of \: Elder\:son} = \dfrac{Age \: of \: Elder\:son}{Father's\:age}$

$\implies \sf \frac{6}{x} = \frac{x}{(72 - x)} \\ \\ \bf on \: cross - multiplying : \\ \\ \implies \sf 6(72 - x) = x {}^{2} \\ \\ \implies \sf 432 - 6x = x {}^{2} \\ \\ \implies \sf {x}^{2} + 6x - 432 = 0$

On splitting it's middle term:

$\implies \sf {x}^{2} + (24 - 18)x - 432 = 0 \\ \\ \implies \sf {x}^{2} + 24x - 18x - 432 = 0 \\ \\ \implies \sf x(x +24) - 18(x + 24) = 0 \\ \\ \implies \sf (x + 24)(x - 18) = 0$

Therefore,

x = 18 or x = (-24)

Since, age cannot be negative. We will neglect it's negative value.

_______________________

Hence, the present age of elder son is 18 years.