A father left all of his money to his 3 children. Child A received 1/6 of his father's money while Child B received 1/7. Child C, who received the remainder of the money , was given $23,490. How much did Child A receive?

First, we need to determine what "slice" of the money Child C received:

Let's call the "slice" Child C received: #c#

100% is the same as 1 so we can write and solve this equation for #c#;

#1/6 + 1/7 + c = 1#

#(7/7 xx 1/6) + (6/6 xx 1/7) + c = 1#

#7/42 + 6/42 + c = 1#

#(7 + 6)/42 + c = 1#

#13/42 + c = 1#

#13/42 - color(red)(13/42) + c = 1 - color(red)(13/42)#

#0 + c = 42/42 - color(red)(13/42)#

#c = (42 - color(red)(13))/42#

#c = 29/42#

Child C received #29/42# of his father's money.

Now, let's call the total amount of money the father left for the 3 children: #a#

We can write this equation and solve for #a#:

#29/42 xx a = $23490#

#color(red)(42)/color(blue)(29) xx 29/42 xx a = color(red)(42)/color(blue)(29) xx $23490#

#cancel(color(red)(42))/cancel(color(blue)(29)) xx color(blue)(cancel(color(black)(29)))/color(red)(cancel(color(black)(42))) xx a = color(red)(42)/color(blue)(29) xx ($810 xx 29)#

#a = color(red)(42)/cancel(color(blue)(29)) xx ($810 xx color(red)(cancel(color(black)(29))))#

#a = color(red)(42) xx $810#

#a = $34020#

We now know the total amount of money the father left to all the children was $34,020.

Child A received #1/6# of this money:

#1/6 xx $34020 =>#

#1/6 xx $5670 xx 6 =>#

#1/color(red)(cancel(color(black)(6))) xx $5670 xx color(red)(cancel(color(black)(6))) =>#

#$5670#

Child A received $5,670