Question

5.
A father is four times as old as his son now. After 24 years he would only be twice as
old as his son. What are the present ages of the father and the son?​

Answer 1

$\large \bold{Answer:}$

$\begin{gathered} \qquad \:\boxed{\begin{aligned}&\:\sf \: Present \: age \: of \: son=12 \: years \qquad \: \\ & \:\sf \: Present \: age \: of \: father=48 \: years\end{aligned}} \qquad \end{gathered}$

$\bold{Step-by-step \: \: explanation:}$

Given :-

• A father is four times as old as his son now.
• After 24 years he would only be twice as old as his son.

To Find :-

• What is the present age of the father and son ?

Solution :-

Let,

$\begin{gathered}\mapsto \bf Present Age_{(Son)} =\: a\: years\\\end{gathered}$

$\begin{gathered}\mapsto \bf Present\: Age_{(Father)} =\: 4a\: years\end{gathered}$

$\bigstar$ Father is four times as old as his son now.

❒ After 24 years their ages will be :

$\begin{gathered}\leadsto \sf Age_{(Son)} =\: (a + 24)\: years\\\end{gathered}$

$\begin{gathered}\leadsto \sf Age_{(Father)} =\: (4a + 24)\: years\end{gathered}$

❒ According to the question :

$\bigstar$ After 24 years he would only be twice as old as his son.

So,

$\begin{gathered}\small \implies \sf\boxed{\bold{\bigg\{Age_{(Father)}\bigg\} =\: 2\bigg\{Age_{(Son)}\bigg\}}}\end{gathered}$

$\begin{gathered}\implies \sf (4a + 24) =\: 2(a + 24)\end{gathered}$

$\begin{gathered}\implies \sf 4a + 24 =\: 2a + 48\end{gathered}$

$\begin{gathered}\implies \sf 4a - 2a =\: 48 - 24\end{gathered}$

$\begin{gathered}\implies \sf 2a =\: 24\end{gathered}$

$\begin{gathered}\implies \sf a =\: \dfrac{\cancel{24}}{\cancel{2}}\end{gathered}$

$\begin{gathered}\implies \sf\bold{a =\: 12}\end{gathered}$

Hence, the required present ages are :

$\dag$ Present Age Of Son :

$\begin{gathered}\dashrightarrow \sf Present\: Age_{(Son)} =\: a\: years\end{gathered}$

$\begin{gathered}\dashrightarrow \sf\bold{\underline{Present\: Age_{(Son)} =\: 12\: years}}\end{gathered}$

$\dag$ Present Age Of Father :

$\begin{gathered}\dashrightarrow \sf Present\: Age_{(Father)} =\: 4a\: years\end{gathered}$

$\begin{gathered}\dashrightarrow \sf Present\: Age_{(Father)} =\: (4 \times 12)\: years\end{gathered}$

$\begin{gathered}\dashrightarrow \sf\bold{\underline{Present\: Age_{(Father)} =\: 48\: years}}\end{gathered}$

$\therefore$ The present age of the father is 48 years and the present age of son is 12 years .

$\rule{170pt}{2pt}$

$\bold{ \pink{ \underline{ \underline {\red{ \boxed{ \purple {\bold{Additional \: \: information:}}}}}}}}$

$\begin{gathered}\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{ \blue{More \: identities}}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

$\bold{ \blue{@TanmayBrainlyTopper}}$