Solution

Amount received by each child

\(\dfrac{\text{Total amount}}{\text{Number of children}}=\dfrac{x^2+12x+35}{x+7}\)

\[ \begin{aligned} \\ &= \frac{ x^2 \ {\color{magenta} + \ 12x} + 35}{(x+7)} \\[6pt] &= \frac{ x^2 \ {\color{magenta} + \ 7x + 5x} + 35}{(x+7)} \\[6pt] &= \frac{ x(x+7) + 5(x+7)}{(x+7)} \\[6pt] &= \frac{ \cancel{(x+7)}(x+5)}{\cancel{(x+7)}} \\[6pt] &= x+5 \end{aligned} \]

Answer Each child received \( \color{red} \text{₹} (x+5) \)

Solution

\[ \begin{aligned} \text{Distance} &= \color{magenta}{2t^4 + 3t^3 - 2t^2 - 9t - 12} \\ \text{Speed} &= \color{magenta}{t^2 - 3} \\[8pt] \text{Time} &=\frac{\text{Distance}}{\text{Speed}} \\[8pt] & =\dfrac{2t^4 + 3t^3 - 2t^2 - 9t - 12}{t^2 - 3} \end{aligned} \] \[ \begin{array}{l} \hspace{1.7cm}\color{green}{ 2t^2 \ + \ 3t \ + \ 4 } \\ t^2 - 3 \enclose{longdiv}{\ \ 2t^4 + 3t^3 - 2t^2 - 9t - 12}\\ \hspace{1.05cm}\overset{\color{red}\scriptsize(-)}{+}\ 2t^4 \hspace{1 cm} \overset{\color{red}\scriptsize(+)}{-} 6t^2 \\ \hline \hspace{1.7cm}0 + 3t^3 + 4t^2 - 9t - 12 \\ \hspace{1.95cm}\overset{\color{red}\scriptsize(-)}{+}\ 3t^3 \hspace{1 cm} \overset{\color{red}\scriptsize(+)}{-} 9t \\ \hline \hspace{2.6cm}0 + 4t^2 + 0t - 12 \\ \hspace{2.8cm}\overset{\color{red}\scriptsize(-)}{+}\ 4t^2 \hspace{1 cm} \overset{\color{red}\scriptsize(+)}{-} 12 \\ \hline \hspace{4.35cm}0 \end{array} \]\[ \begin{aligned} \color{green}{\text{Quotient}} &= 2t^2 + 3t + 4 \\ \color{green}{\text{Remainder}} &= 0 \end{aligned} \]

Answer Time taken \(=\ \color{red}{(2t^2 + 3t + 4) \text{ hours}}\)