One dime is worth 2 nickels, so having 7 times as many dimes as nickels means that the total value of the dimes is (7x2) x the total value of the nickels. So, his dimes are worth 14 times more than his nickels, and are worth $3.90 more than the nickels. So, if the value of nickels is n, and the value of dimes is d, n+3.90=d, and 14n=d. This means that 14n=n+3.90
14n is the same as saying n+13n, so "n+13n=n+3.90"
Cancel "n" from both sides, and you get "13n=3.90".
Divide out 13, and you get "n=0.30". Your total value of nickels is 30 cents, so you must have 6 nickels. Seven times as many dimes leaves you with 6 nickels and 42 dimes.