The ratio of the number of boys to the number of girls in an auditorium was 6:7  If $\frac{2}{3}$  of the boys left the auditorium, there would be 70 more girls than boys in the auditorium. How many students were in the auditorium at first?

Number of boys : number of girls  = 6:7

  There fore number of boys =6x
And number of girls  =7x
Given , $\frac{2}{3}$of boys leave the auditorium.
  Number of boys $6\mathrm{x}-\frac{2}{3}\times 6\mathrm{x}$
$=6\mathrm{x}-4\mathrm{x}=2\mathrm{x}$
According to the question,
$7\mathrm{x}-70=2\mathrm{x}$
$$\begin{gathered} \Rightarrow 5\mathrm{x}=70 \\ \Rightarrow \mathrm{x}=14 \end{gathered}$$
So, number of students  $=6\mathrm{x}+7\mathrm{x}$
$=13\mathrm{x}=13\times 14=182$