A jar of coins contains nickels, dimes, and quarters

A jar of coins contains nickels, dimes, and quarters. The total number of coins is 12 and the total value is $2.50. How many of each coin are there?

The correct answer and explanation is:

To find the number of each type of coin in the jar, we define:

• Let n be the number of nickels
• Let d be the number of dimes
• Let q be the number of quarters

We are given two pieces of information:

1. The total number of coins is 12:
   n + d + q = 12
2. The total value is $2.50, which is 250 cents:
   5n + 10d + 25q = 250

To solve this, we can simplify the second equation by dividing every term by 5:
n + 2d + 5q = 50

Now we have a system of two equations:

1. n + d + q = 12
2. n + 2d + 5q = 50

We subtract the first equation from the second:
(n + 2d + 5q) – (n + d + q) = 50 – 12
n cancels out:
(2d – d) + (5q – q) = 38
d + 4q = 38

Now solve for d:
d = 38 – 4q

Now substitute back into the first equation:
n + d + q = 12
n + (38 – 4q) + q = 12
n = 12 – (38 – 4q + q)
n = 12 – (38 – 3q)
n = -26 + 3q

Now test small integer values for q (since q must be a whole number):

• If q = 4:
  d = 38 – 4(4) = 22
  n = -26 + 3(4) = -14 (not valid)
• If q = 5:
  d = 38 – 20 = 18
  n = -26 + 15 = -11 (not valid)
• If q = 6:
  d = 38 – 24 = 14
  n = -26 + 18 = -8 (not valid)
• If q = 7:
  d = 38 – 28 = 10
  n = -26 + 21 = -5 (not valid)
• If q = 8:
  d = 38 – 32 = 6
  n = -26 + 24 = -2 (still invalid)
• If q = 9:
  d = 38 – 36 = 2
  n = -26 + 27 = 1 → valid

Now check if n + d + q = 1 + 2 + 9 = 12 ✔
Check value:
5(1) + 10(2) + 25(9) = 5 + 20 + 225 = 250 cents ✔

Answer:
Nickels: 1
Dimes: 2
Quarters: 9

This is the only combination of non-negative integers that satisfies both conditions: a total of 12 coins and a value of $2.50.