Vesna is the banker in a board game that uses \(\$1\), \(\$10\), and \(\$100\) bills.
Vesna needs to give a player \(\$2163\). How can you do this using the fewest total number of bills? How can you do this using the greatest total number of bills?
Is it possible to give a player \(\$254\) using exactly \(20\) bills in total? How about using exactly \(30\) bills in total? If so, show how it’s possible. If not, explain why it’s not possible.
Extension: Vesna likes when she can give a player the same number of each type of bill. For which total amounts of money is this possible? Explain.
To give a player \(\$2163\) using the fewest total number of bills, we first use as many \(\$100\) bills as we can. Since the number \(2163\) has \(21\) hundreds, then we can use at most \(21\) of the \(\$100\) bills. This gives \(21 \times \$100 = \$2100\), so we are left with \(\$2163 - \$2100=\$63\). Next we use as many \(\$10\) bills as we can. Since the number \(63\) has \(6\) tens, then we can use at most \(6\) of the \(\$10\) bills. This gives \(6 \times \$10=\$60\). We are then left with \(\$3\), so we need \(3\) of the \(\$1\) bills. Thus in total, we use: \[21 \times \$100\text{ bills}; \quad 6 \times \$10\text{ bills}; \quad 3 \times \$1\text{ bills}\] This is a total of \(21+6+3=30\) bills.
To give a player \(\$2163\) using the greatest total number of bills, we want to use as many \(\$1\) bills as possible. If we use all \(\$1\) bills, then we will use \(2163\) bills in total.
Using the strategy from (a) to use the fewest total number of bills, we can give \(\$254\) as follows: \[2 \times \$100\text{ bills}; \quad 5 \times \$10\text{ bills}; \quad 4 \times \$1\text{ bills}\] This uses a total of \(2+5+4=11\) bills. Let’s try replacing one \(\$100\) bill with ten \(\$10\) bills. This gives: \[1 \times \$100\text{ bill}; \quad 15 \times \$10\text{ bills}; \quad 4 \times \$1\text{ bills}\] This uses a total of \(1+15+4=20\) bills, so it is possible to give a player \(\$254\) using exactly \(20\) bills in total.
Notice that every time we replace a bill with ten smaller bills, the total number of bills increases by \(9\). This is true if we replace one \(\$100\) bill with ten \(\$10\) bills or if we replace one \(\$10\) bill with ten \(\$1\) bills. So the total number of bills is a sequence that starts at \(11\) and increases by \(9\) each time, until it reaches \(254\) (which is the greatest total number of bills that can be used for \(\$254\)). Writing out more terms in the sequence gives \(11\), \(20\), \(29\), \(38\), \(\ldots\). Since \(30\) is not in this sequence, we can not give a player \(\$254\) using exactly \(30\) bills in total.
Solution to Extension: If a player gets \(1\) of each bill, then the total amount is \(1 \times \$100 + 1 \times \$10 + 1\times \$1=\$111\). If a player gets \(2\) of each bill, then the total amount will be \(2 \times \$111 = \$222\), because they have twice as many of each bill, so the total amount will double. Similarly, if a player gets \(6\) of each bill, then the total amount will be \(6 \times \$111 = \$666\), and if a player gets \(15\) of each bill, then the total amount will be \(15 \times \$111 = \$1665\). Thus, the total amount will always be a multiple of \(111\).
We can also use variables to explain this. Suppose a player receives \(n\) bills of each type. Then the total amount is equal to \((n \times 100 + n \times 10 + n \times 1)\). This is the same as \(n \times (100+10+1)\), which equals \(n \times 111\). Therefore, the total amount must be a multiple of \(111\).