Counting Bills and Coins, and Calculating Change - Third Grade Mathematics

Money is one of the most practical applications of mathematics in the real world. Whenever you purchase snacks at a market, save allowance in a piggy bank, or pay for admission to a movie, you are counting bills and coins and calculating change. In this chapter, you will master determining the total value of mixed collections of currency, writing money amounts with dollar signs and decimals, and calculating change using counting-on and subtraction strategies.

Values of Coins and Dollar Bills

Our currency system includes coins and paper bills, each with a specific value in cents or dollars:

Coin / Bill     Value in Cents    Decimal Value
Penny           1 cent            0.01 dollars
Nickel          5 cents           0.05 dollars
Dime            10 cents          0.10 dollars
Quarter         25 cents          0.25 dollars
Half-Dollar     50 cents          0.50 dollars
One-Dollar Bill 100 cents         1.00 dollar
Five-Dollar Bill 500 cents        5.00 dollars
Ten-Dollar Bill 1,000 cents       10.00 dollars
Twenty-Dollar Bill 2,000 cents    20.00 dollars

Notice the key coin relationships:
- 100 pennies = 1 dollar

- 20 nickels = 1 dollar

- 10 dimes = 1 dollar

- 4 quarters = 1 dollar


Strategy for Counting Mixed Collections of Money

To count a mixed pile of bills and coins without making mistakes, follow this systematic strategy:
1. Sort coins and bills by denomination from greatest value to least value:
Bills first (20s, 10s, 5s, 1s), then quarters, dimes, nickels, and pennies.
2. Count on by adding each value:
Start with the largest value and keep a running total.

Example: You have 1 five-dollar bill, 2 one-dollar bills, 3 quarters, 1 dime, and 2 pennies.

Count Bills:
5.00 dollars ... + 1.00 dollar = 6.00 dollars ... + 1.00 dollar = 7.00 dollars

Count Coins (Quarters first):
7.00 dollars + 25 cents = 7.25 dollars
7.25 dollars + 25 cents = 7.50 dollars
7.50 dollars + 25 cents = 7.75 dollars

Count Dimes:
7.75 dollars + 10 cents = 7.85 dollars

Count Pennies:
7.85 dollars + 1 cent = 7.86 dollars
7.86 dollars + 1 cent = 7.87 dollars

Total Value: 7.87 dollars (seven dollars and eighty-seven cents)

How to Calculate Change

When you pay for an item with more money than the item costs, the cashier returns the difference to you as change.

Change = Amount Paid - Cost of Item

There are two primary methods for finding change:

Method 1: The "Count Up" Method (Cashier's Strategy)

Suppose you buy a comic book for 3 dollars and 65 cents (3.65 dollars) and pay with a 5-dollar bill. Start at the cost (3.65 dollars) and count up to the amount paid (5.00 dollars):
- Add 5 cents (1 nickel) to reach 3.70 dollars.

- Add 30 cents (3 dimes) to reach 4.00 dollars.

- Add 1.00 dollar (1 dollar bill) to reach 5.00 dollars.
Total change: 1.00 dollar + 30 cents + 5 cents = 1 dollar and 35 cents (1.35 dollars)!

Method 2: Vertical Subtraction (Across Zeros)

You can also subtract vertically, treating dollars and cents as whole cents:
5.00 dollars is 500 cents; 3.65 dollars is 365 cents.


    4  9 10
    5  0  0   cents
  - 3  6  5   cents
  ---------
    1  3  5   cents = 1 dollar and 35 cents (1.35 dollars)

Both methods lead to the exact same result: 1 dollar and 35 cents in change!

Chapter Practice Exercises

  1. Find the total value of each coin collection: a. 3 quarters, 2 dimes, 1 nickel b. 2 quarters, 3 dimes, 4 pennies c. 4 quarters, 1 half-dollar, 2 dimes
  2. Write the total amount in words and as a decimal amount of dollars: a. 2 five-dollar bills, 3 one-dollar bills, 2 quarters, 1 dime b. 1 ten-dollar bill, 1 five-dollar bill, 1 quarter, 3 nickels, 4 pennies
  3. Calculate the change for each purchase: a. Item cost: 2.40 dollars | Paid with: 5.00 dollar bill b. Item cost: 7.25 dollars | Paid with: 10.00 dollar bill c. Item cost: 14.60 dollars | Paid with: 20.00 dollar bill
  4. Liam buys a toy airplane that costs 4.18 dollars. He hands the cashier a 5-dollar bill. What coins and bills could the cashier give Liam as his exact change?
  5. Maya has 8 dollars and 50 cents. She wants to buy a book that costs 6 dollars and 75 cents and a bookmark that costs 1 dollar and 25 cents. Does Maya have enough money? Show your addition work.

Solutions and Step-by-Step Answers

  1. Coin totals: a. 3 quarters (75 cents) + 2 dimes (20 cents) + 1 nickel (5 cents) = 75 + 20 + 5 = 100 cents = 1.00 dollar. b. 2 quarters (50 cents) + 3 dimes (30 cents) + 4 pennies (4 cents) = 50 + 30 + 4 = 84 cents = 0.84 dollars. c. 4 quarters (1.00 dollar) + 1 half-dollar (50 cents) + 2 dimes (20 cents) = 1.00 dollar + 70 cents = 1.70 dollars.
  2. Dollar amounts: a. Bills: 2 fives (10 dollars) + 3 ones (3 dollars) = 13 dollars. Coins: 2 quarters (50 cents) + 1 dime (10 cents) = 60 cents. Total: 13.60 dollars (thirteen dollars and sixty cents). b. Bills: 10 + 5 = 15 dollars. Coins: 25 cents + 15 cents + 4 cents = 44 cents. Total: 15.44 dollars (fifteen dollars and forty-four cents).
  3. Calculating change: a. 5.00 - 2.40 = 2.60 dollars change (2 dollars and 60 cents). b. 10.00 - 7.25 = 2.75 dollars change (2 dollars and 75 cents). c. 20.00 - 14.60 = 5.40 dollars change (5 dollars and 40 cents).
  4. Liam's change: 5.00 - 4.18 = 0.82 dollars (82 cents). Possible coin combination: 3 quarters (75 cents), 1 nickel (5 cents), and 2 pennies (2 cents) -> 75 + 5 + 2 = 82 cents.
  5. Maya's purchases: 6.75 + 1.25 = 8.00 dollars total cost. Maya has 8.50 dollars. Since 8.50 > 8.00, Maya has enough money! She will have 50 cents (0.50 dollars) left over.