let

x = the number of girls

y = the number of boys

2/7 of the number of girls: (2/7)x

3/5 of the number of boys: (3/5)y

at a festival 2/7 of the number of girls was equal to 3/5 of the number of boys:

(2/7)x = (3/5)y

there were 165 fewer boys than girls:

y = x - 165

by solving the system of equations:

(2/7)x = (3/5)y

y = x - 165

we find:

x = 315 girls

y = 150 boys

click here to see the step by step solution of the system of equations:

Click to see all the steps

x + y = 315 + 150 = 465 children

there were 465 children in all.