Michael has some coins in his pocket consisting of dimes, nickels, and pennies. He has two more nickels than dimes, and three times as many pennies as nickels. How many of each kind of coins does he have if the total value is 52 cents?

X = number of dimes

x + 2 = number of nickels

3 (x + 2) = number of pennies

10x + 5 (x + 2) + 3 (x + 2) = 52

10x + 5x + 10 + 3x + 6 = 52

18x = 36

x = 2

x = 2 (number of dimes)

x + 2 = 4 (number of nickels)

3 (x + 2) = 12 (number of pennies)

Check:

2 (10) = 20 cents in dimes

4 (5) = 20 cents in nickels

12 (1) = 12 cents in pennies

Total = 52 cents

He has d dimes.

He has 2 more nickles than dimes, so he has d + 2 nickles.

He has 3 times as many pennies as nickles. He has 3 (d + 2) pennies.

Now we figure out the value of each type of coin in cents.

d dimes are worth 10d cents.

d + 2 nickels are worth 5 (d + 2) cents.

3 (d + 2) pennies are worth 3 (d + 2) cents.

Add all expressions of value and set equal to 52 cents.

10d + 5 (d + 2) + 3 (d + 2) = 52

10d + 5d + 10 + 3d + 6 = 52

18d + 16 = 52

18d = 36

d = 2

He has 2 dimes.

He has 2 more nickles than dimes, so he has 2 + 2 nickles, or 4 nickles.

He has three times as many pennies as nickels, or 3 (4) = 12 pennies.

Answer: 2 dimes, 4 nickles, 12 pennies.