How many ways can you get $1.05 using only nickels, dimes, and quarters
I would create a table of values using
quarters, nickels and dimes
Q D N
4 0 1
3 3 0
3 2 2
3 1 4
3 0 6 Did you notice the pattern? take away 1 dime, add 2 nickels
2 5 1
2 4 3
2 3 5
2 2 7
2 1 9
2 0 11
1 8 0
1 7 2
...
1 0 16
0 10 1
0 9 3
0 8 5
...
0 0 21
Can you complete the chart and count the triples?
To find the number of ways to get $1.05 using only nickels, dimes, and quarters, we can break it down into different cases.
1. Using only quarters: Since each quarter is worth $0.25, we can use 4 quarters to make $1.00. Then, we need an additional nickel to reach $1.05. Therefore, there is only one way to use only quarters in this case.
2. Using quarters and dimes: If we use 3 quarters (worth $0.75), we still need $0.30 to reach $1.05. We can achieve this by using 3 dimes ($0.30). Hence, there is only one way to use quarters and dimes.
3. Using quarters, dimes, and nickels: If we don't use any quarters, we need to find combinations of dimes and nickels that add up to $1.05. Starting with 10 dimes ($1.00), we can subtract it in combinations with nickels to reach $1.05. Here are the possible combinations:
- 10 dimes and 1 nickel
- 9 dimes and 3 nickels
- 8 dimes and 5 nickels
- 7 dimes and 7 nickels
- 6 dimes and 9 nickels
- 5 dimes and 11 nickels
- 4 dimes and 13 nickels
- 3 dimes and 15 nickels
- 2 dimes and 17 nickels
- 1 dime and 19 nickels
- 21 nickels
In total, there are 11 combinations for this case.
Thus, the total number of ways to get $1.05 using only nickels, dimes, and quarters is 1 (using only quarters) + 1 (using quarters and dimes) + 11 (using quarters, dimes, and nickels) = 13 ways.