Solution

\[ \begin{aligned} \text{Let the number be } &= x \\[6pt] \text{half the number} &= \frac{x}{2} \\[6pt] \text{one-third of the number} &= \frac{x}{3} \\[6pt] \text{one-sixth of the number} &= \frac{x}{6} \\[6pt] \text{Average} &= \frac{\frac{x}{2} + \frac{x}{3} + \frac{x}{6}}{3} \\[6pt] &= \frac{\frac{3x + 2x + x}{6}}{3} \\[6pt] &= \frac{\frac{6x}{6}}{3} \\[6pt] \text{Average}&= \frac{x}{3} \\[6pt] \textbf{According to the} & \textbf{ question} \\[6pt] x - \text{(Average)} &= 6 \\[6pt] x - \frac{x}{3} &= 6 \\[6pt] \frac{3x - x}{3} &= 6 \\[6pt] \frac{2x}{3} &= 6 \\[6pt] 2x &= 18 \\[6pt] \color{green} x &= \color{green} 9 \end{aligned} \]

AnswerThe number is \(\color{red}{ 9 }\)

Solution

\[ \begin{aligned} \frac{x^2 - 3x - 28}{x^2 - 49} &= \frac{3}{17} \\[6pt] \frac{x^2 - 7x + 4x - 28}{x^2 - 7^2} &= \frac{3}{17} \\[6pt] \frac{x(x - 7) + 4(x - 7)}{(x + 7)(x - 7)} &= \frac{3}{17} \\[6pt] \frac{(x - 7)(x + 4)}{(x + 7)(x - 7)} &= \frac{3}{17} \\[6pt]\frac{x + 4}{x + 7} &= \frac{3}{17} \\[6pt] 17(x + 4) &= 3(x + 7) \\[6pt] 17x + 68 &= 3x + 21 \\[6pt] 17x - 3x &= 21 - 68 \\[6pt] 14x &= -47 \\[6pt] \color{green} x &= \color{green} \frac{-47}{14} \end{aligned} \]

Answer \(\color{red}{ x = \dfrac{-47}{14} }\)