Tammy wants to get change for 30 cents. The only coins she can get are quarters, nickels, and dimes. How many different ways can she get 30 cents using only these coins?
Just list them:
Q D N
1 0 1
0 3 0
0 2 2
0 1 4
0 0 6
To find the different ways Tammy can get 30 cents using only quarters, nickels, and dimes, we can use a systematic approach.
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.
To find the different ways Tammy can get change for 30 cents, we can use a combination of quarters, nickels, and dimes.
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.
qn
ddd
now replace each d with nn for the others.