Question

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

Answer 1

AnswEr :

• Age of younger son = 6 years
• Sum of ages of elder son and father ages is 72 years.
• Ages of younger son, elder son and father are continuous proportion.
• Find the Age of Elder Son.

Let the Age of Elder Son be n years.

⋆ A. T. Q.

↠ (Elder Son + Father)'s Age = 72 years

↠ n years + Father's Age = 72 years

↠ Father's Age = (72 – n) years

$\rule{100}{2}$

• According to the Question Now :

$\longrightarrow \texttt{Younger Son : Elder Son :: Elder Son : Father} \\\\\longrightarrow \tt{\dfrac{Younger \:Son's\:Age}{Elder \:Son's\:Age} = \dfrac{Elder \:Son's\:Age}{Father's\:Age} } \\ \\\longrightarrow \tt \dfrac{6}{n} =\dfrac{n}{(72 - n)} \\ \\\longrightarrow \tt6(72 - n) = n \times n \\ \\\longrightarrow \tt432 - 6n ={n}^{2} \\ \\\longrightarrow \tt {n}^{2} + 6n - 432 = 0 \\ \\\scriptsize\star\quad\text{Using Discriminat Formula Here :} \\ \\\longrightarrow \tt n = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac}}{2a} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm \sqrt{ {(6)}^{2} - (4 \times 1 \times - 432)}}{2 \times 1} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm \sqrt{36 + 1728} }{2} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm \sqrt{1764} }{2} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm \sqrt{7 \times 7 \times 6 \times 6} }{2} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm(7 \times 6)}{2} \\ \\\longrightarrow \tt n = \dfrac{ - 6 \pm42}{2} \\ \\\longrightarrow \tt n = \dfrac{ - 6 + 42}{2} \quad or \quad n = \dfrac{ - 6 - 42}{2} \\ \\\longrightarrow \tt n = \dfrac{36}{2} \quad or \quad n = \dfrac{ - 48}{2} \\ \\\longrightarrow \green{\tt n = 18 \:years}\quad \tt or \quad \red{n = - 24 \:years}$

Since Age can't be Negative, that's why we will neglect Negative Value.

⋆ Elder Son = n years = 18 years

⠀

∴ Therefore, Elder Son is 18 years old.

Answer 2

Answer:

Let the Age of Elder Son be a.

= > 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 years.