qn
ddd
now replace each d with nn for the others.
ddd
now replace each d with nn for the others.
Q D N
1 0 1
0 3 0
0 2 2
0 1 4
0 0 6
Let's break down the problem by considering each type of coin separately.
1. Quarters: Since the value of a quarter is 25 cents, Tammy can use at most one quarter to make 30 cents. So, there are two possible scenarios:
a) Tammy uses 0 quarters (0*25 = 0 cents) and needs to make 30 cents using nickels and dimes.
b) Tammy uses 1 quarter (1*25 = 25 cents) and needs to make 5 cents using nickels and dimes.
2. Nickels: Since the value of a nickel is 5 cents, Tammy can use at most six nickels to make 30 cents. So, there are seven possible scenarios:
a) Tammy uses 0 nickels (0*5 = 0 cents) and needs to make 30 cents using dimes.
b) Tammy uses 1 nickel (1*5 = 5 cents) and needs to make 25 cents using dimes.
c) Tammy uses 2 nickels (2*5 = 10 cents) and needs to make 20 cents using dimes.
...
f) Tammy uses 5 nickels (5*5 = 25 cents) and needs to make 5 cents using dimes.
g) Tammy uses 6 nickels (6*5 = 30 cents) and doesn't need any dimes.
3. Dimes: Since the value of a dime is 10 cents, Tammy can use up to three dimes to make 30 cents. So, there are four possible scenarios:
a) Tammy uses 0 dimes (0*10 = 0 cents) and needs to make 30 cents using nickels.
b) Tammy uses 1 dime (1*10 = 10 cents) and needs to make 20 cents using nickels.
c) Tammy uses 2 dimes (2*10 = 20 cents) and needs to make 10 cents using nickels.
d) Tammy uses 3 dimes (3*10 = 30 cents) and doesn't need any nickels.
Now we can calculate the total number of different ways Tammy can get 30 cents. We add up the number of possibilities for each scenario:
For the scenario with 0 quarters:
- 7 possibilities for nickels (0-6).
- 4 possibilities for dimes (0-3).
Total: 7 x 4 = 28 possibilities.
For the scenario with 1 quarter:
- 2 possibilities for nickels (0-1).
- 4 possibilities for dimes (0-3).
Total: 2 x 4 = 8 possibilities.
Adding these together, Tammy can get change for 30 cents in a total of 28 + 8 = 36 different ways.
Let's break down our problem into smaller sub-problems.
1. Using quarters: Since quarters are worth 25 cents, Tammy can use either 0, 1, or 2 quarters.
a. If Tammy uses 0 quarters, she would have to make 30 cents using only nickels and dimes. We'll come back to this later.
b. If Tammy uses 1 quarter, she would have 30 - 25 = 5 cents left. Tammy can use nickels and dimes to make 5 cents.
i. If Tammy uses 0 nickels, she would have 5 cents left. Tammy can use only dimes to make 5 cents, which is a single option.
ii. If Tammy uses 1 nickel, she would have 5 - 5 = 0 cents left. Tammy has used 1 nickel and 1 quarter, then she has reached a total of 30 cents. This is another option.
c. If Tammy uses 2 quarters, she would have 30 - 2 * 25 = 30 - 50 = -20 cents left. Since Tammy cannot have a negative total, this scenario is not valid.
2. Using no quarters: As mentioned earlier, if Tammy doesn't use any quarters, she needs to make 30 cents using only nickels and dimes.
a. If Tammy uses 0 nickels, she would have 30 cents left to make. Tammy can use dimes to make the remaining 30 cents. This is a valid option.
b. If Tammy uses 1 nickel, she would have 30 - 5 = 25 cents left. Tammy can use either 0 or 1 dime to make the remaining 25 cents.
i. If Tammy uses 0 dimes, she would have 25 cents left. Tammy can use only nickels to make the remaining 25 cents. This is a valid option.
ii. If Tammy uses 1 dime, she would have 25 - 10 = 15 cents left. Tammy can use only nickels to make the remaining 15 cents. This is another valid option.
c. If Tammy uses 2 nickels, she would have 30 - 2 * 5 = 20 cents left. Tammy can use either 0, 1, or 2 dimes to make the remaining 20 cents.
i. If Tammy uses 0 dimes, she would have 20 cents left. Tammy can use only nickels to make the remaining 20 cents. This is another valid option.
ii. If Tammy uses 1 dime, she would have 20 - 10 = 10 cents left. Tammy can use either 0 or 1 nickel to make the remaining 10 cents.
- If Tammy uses 0 nickels, she would have 10 cents left. Tammy can use only dimes to make the remaining 10 cents. This is a valid option.
- If Tammy uses 1 nickel, she would have 10 - 5 = 5 cents left. Tammy can use only dimes to make the remaining 5 cents. This is another valid option.
iii. If Tammy uses 2 dimes, she would have 20 - 2 * 10 = 0 cents left. Tammy has used 2 dimes and 2 nickels, then she has reached a total of 30 cents. This is another valid option.
Summarizing our findings, Tammy can get change for 30 cents in the following ways:
1. Using 0 quarters: 1 option
2. Using 1 quarter and 0 nickels: 1 option
3. Using 0 quarters and 0 nickels: 1 option
4. Using 0 quarters and 1 nickel: 1 option
5. Using 0 quarters, 2 nickels, and 0 dimes: 1 option
6. Using 0 quarters, 2 nickels, and 1 dime: 1 option
Therefore, there are a total of 6 different ways Tammy can get change for 30 cents using only quarters, nickels, and dimes.