Question

the difference of the ages of son and father is 28 years the difference of square of their ages is 1848.Find the ages of son and father​

Answer 1

$\huge\orange{\boxed{\green{\mathbb{\underbrace{\overbrace{\fcolorbox{blue}{navy}{\underline{\color{ivory}{꧁Let us ꧂ }}}}}}}}}$${navy}{\underline{\color{ivory}{꧁Consider꧂ }}}}}}}}}$

❍ Let's Consider the age of father and son be a yrs & b yrs , respectively.

      $\begin{gathered}{ \underline{ \fbox{ \textsf{ \textbf{ \large{ \color{green}{Case 1}}}}}}} \end{gathered}$

⠀⠀⠀The difference of the ages of son and father is 28 years .

$\\\\huge\mathcal\blue{\qquad :\implies \sf Age \:of \:Father \: - \: Age \:of \: Son \: = 28 \: }$

$\huge\mathcal\green{\qquad :\implies \sf a \: - \: b \: = 28 \: \\\}$

${\huge{\underline{\bf{\pink{\qquad :\implies \bf a \: = \: 28 + b \:\qquad \qquad \bigg\lgroup \sf{ Equation \: 1 \:\:}\bigg\rgroup\\\}}}}}$

                      ${ \textsf{ \textbf{ \large{ \sf{ \color{red}{➣ Case 2 }}}}}}$

$\huge\mathfrak\red{\qquad :\implies \sf ( Age \:of \:Father)^2 \: - \: (Age \:of \: Son)^2 \: = 28 \:}$

$\huge\mathcal\blue{\qquad :\implies \sf ( a)^2 \: - \: (b)^2 \: = 28 \: \\\}$

$\begin{gathered}\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: Eq.1 \:\: \::}}\\\end{gathered}$

${\huge{\underline{\bf{\pink{\qquad :\implies \sf ( a)^2 \: - \: (b)^2 \: = 28 \:}}}}}$

${\huge{\boxed{\sf{\green{\qquad :\implies \sf ( 28 + b )^2 \: - \: (b)^2 \: = 28 \: }}}}}$

$\begin{gathered}\dag\:\:\it{ As,\:We\:know\:that\::}\\\end{gathered}$

$\huge\mathfrak\red{\qquad \dag\:\:\bigg\lgroup \sf{ Algebraic \:Indentity \:\:\:: (a + b)^2 = a^2 + b^2 + 2ab }$

$\begin{gathered}\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \: Algebraic \: Indentity \::}}\\\end{gathered}$

$\huge\mathcal\blue{\qquad :\implies \sf ( 28 + b )^2 \: - \: (b)^2 \: = 1848 \: }$

${\huge{\underline{\bf{\pink{\qquad :\implies \sf 28^2 + b^2 + 2 \times 28 \times b \: - \: (b)^2 \: = 1848 \: \\\}}}}}$

${\huge{\boxed{\sf{\green{\qquad :\implies \sf 28^2 \cancel{+ b^2} + 2 \times 28 \times b \: \cancel{- \: (b)^2} \: = 1848\: }}}}}$$\begin{gathered}\qquad :\implies \sf 28^2 + 2 \times 28 \times b \: \: = 1848 \: \\\end{gathered}$

$\begin{gathered}\qquad :\implies \sf 784 + 2 \times 28 \times b \: \: = 1848 \: \\\end{gathered}$

$\huge\mathfrak\red{quad :\implies \sf 784 + 56b \: \: = 1848 \: \\}$

$\huge\mathcal\blue{\qquad :\implies \sf 56b \: \: = 1848 - 784\:}$

${\huge{\underline{\bf{\pink{\qquad :\implies \sf 56b \: \: = 1064\}}}}}$

$\begin{gathered}\qquad :\implies \sf b \: \: = \cancel{\dfrac{1064}{56}}\: \\\end{gathered}$

$\begin{gathered}\qquad :\implies \frak{\underline{\purple{\:b= 19 \:yrs }} }\bigstar \\\end{gathered}$

$\begin{gathered}\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \:value \:of \:b \:in \:Eq.1 \: \:\: \::}}\\\end{gathered}$

$\begin{gathered}\qquad :\implies \bf a \: = \: 28 + b \:\qquad \qquad \bigg\lgroup \sf{ Equation \: 1 \:\:}\bigg\rgroup\\\end{gathered}$

$\begin{gathered}\qquad :\implies \frak{\underline{\purple{\:a = 47 \:yrs }} }\bigstar \\\end{gathered}$

Therefore,

• Son's age is : b = 19 yrs
• Father's age is : a = 47 yrs

$⠀\begin{gathered}\therefore {\underline{ \sf \:Hence,\:The \:age \:of\:Son \:and \:Father \:are\:\bf 19 \:yrs \: \& \: 47 \:yrs \: \sf , respectively . \:}}\\\end{gathered}$