Answer:
[tex]\boxed{365}[/tex]
Step-by-step explanation:
Let g = number of girls
and b = number of boys
We have conditions (1) and (2):
[tex]\begin{array}{lrcll}(1) &\frac{2}{7}g & = & \frac{3}{5}b & \\(2) & g - b & = & 165 &\\(3) & 10g & = & 21b & \text{Multiplied each side of (1) by lcm of denominators}\\(4)& g & = & 165 + b &\text{Added b to each side of (2)}\\ & 10(165 + b) & = & 21b & \text{Substituted 4 into (3)} \\\end{array}[/tex]
[tex]\begin{array}{lrcll} & 1650 + 10b & = & 21b & \text{Distributed the 10} \\ & 1650 & = & 11b & \text{Subtracted 10b from each side} \\ (5) & b & = & 150 &\text{Divided each side by 11} \\ & g - 150 & = & 165 & \text{Substituted (5) into (2)} \\ & g & = & 215 &\text{Added 150 to each side} \\\\ & g + b & = & 365 &\text{Added girls and boys} \\\end{array}[/tex]
[tex]\text{The number of children at the festival was \boxed{\textbf{365}}}[/tex]
Check:
[tex]\begin{array}{rlcrl}\frac{2}{7}\times315& = \frac{3}{5} \times150 & \qquad & 315 - 160 & =165\\90 & = 90& \qquad & 165 & = 165\end{array}[/tex]
