Ans: (a) $6:7:17$ (b) $\dfrac{43}{50}$
- Consider the equation $7a=6b$, we have
$\begin{array}{rcl}
7a & = & 6b \\
\dfrac{a}{b} & = & \dfrac{6}{7} \\
a:b & = & 6:7
\end{array}$Then let $a=6k$ and $b=7k$, where $k\neq 0$. Consider the equation $\dfrac{4a-3c}{2b-c}=9$, we have
$\begin{array}{rcl}
\dfrac{4a-3c}{2b-c} & = & 9 \\
\dfrac{4(6k)-3c}{2(7k)-c} & = & 9 \\
24k-3c & = & 9(14k-c) \\
24k-3c & = & 126k-9c \\
6c & = & 102k \\
c & = & 17k
\end{array}$Hence, we have
$\begin{array}{cl}
& a:b:c \\
= & 6k:7k:17k \\
= & 6:7:17
\end{array}$ - For $a=6k$, $b=7k$ and $c=17k$, we have
$\begin{array}{cl}
& \dfrac{5a+8b}{7b+3c} \\
= & \dfrac{5(6k)+8(7k)}{7(7k)+3(17k)} \\
= & \dfrac{86k}{100k} \\
= & \dfrac{43}{50}
\end{array}$
