Adding and Subtracting Money by Denomination

Property To find the total value or the difference between two sets of money, group the bills and coins by their denomination. Calculate the total value for each denomination separately, then sum these values for the final amount. For subtraction, you may need to make change from a larger denomination.

Examples Addition: You have two $\$10$ bills, one $\$5$ bill, and 3 quarters. Your friend has one $\$10$ bill, two $\$5$ bills, and 2 quarters. To find the total, combine like denominations: $$ (2+1) \times \$10.00 + (1+2) \times \$5.00 + (3+2) \times \$0.25 $$ $$ 3 \times \$10.00 + 3 \times \$5.00 + 5 \times \$0.25 = \$30.00 + \$15.00 + \$0.25 + \$0.25+ \$0.25+ \$0.25+ \$0.25 = \$46.25 $$ Subtraction: You have one $\$20$ bill and want to buy an item that costs $\$12.50$. You can think of the $\$20$ bill as one $\$10$ bill, one $\$5$ bill, and five $\$1$ bills (or other combinations). To find the change from $\$12.50$ ($\$10 + \$2 + 2$ quarters): $$ \$20.00 \$12.50 = (\$10 + \$5 + \$1 + \$1 + \$1 + \$1 + \$1) (\$10 + \$2 + \$0.25 + \$0.25) = \$7.50 $$ Your change could be one $\$5$ bill, two $\$1$ bills, and 2 quarters.

Explanation This method mirrors how we handle physical cash by grouping similar bills and coins together. First, count the number of each denomination (e.g., all the ten dollar bills, all the quarters). Then, find the total value for each group and add these values together to get the grand total. This strategy helps reinforce the value of each bill and coin and provides a practical way to perform calculations without solely relying on decimal algorithms.