Let 'x' be the number of nickles and 'y' be the number of dimes. Also x and y are non-negative integer numbers. We need to find x and y.

We know that

1 nickle = 5 cents

1 dime = 10 cents

$1 = 100 cents

Following the conditions we have

x + y = 30 (a box contains 30 coins in nickles and dimes)

5x + 10 y = 240 ( all together worth $2.40 = 2.40 * 100 cents = 240 cents)

By solving this system we get the solution

5 (30 - y) + 10 y = 240

5y = 90

y = 18 dimes

x = 30 - y

x = 12 nickles

There are 12 nickles and 18 dimes in the box.