John received change worth $13. He received 10 more dimes than nickels and 22 more quarters than dimes. How many coins of each did he receive?

If the amount of money in dimes is 10d (since there are 10 cents per dime and with d being the number of dimes), the amount of money in nickels is 5n and the amount in quarters is 22q (if n=the number of nickels and q=the number of quarters), we can write 10d+5n+25q=1300 (since 1300 cents=13 dollars). In addition, the number of nickels is d-10 (since n+10=d, subtract 10 from both sides to get d-10=n) and the number of quarters is 22+d. Substituting those in, 10d+5 (d-10) + 25 (d+22) = 10d+5d-50+25d+550=40d+500=1300. Subtracting 500 from both sides, we get 800=40d. Next, we divide both sides by 40 to get d=800/40=20. Next, since n=d-10, 20-10=10=n. In addition, since d+22=q, 20+22=42=q.