Coin Combinations for 65¢ with One Quarter

Tawny has 65¢ in her pocket. She has only one quarter and no pennies. What different coin combinations might she have?

Tawny has 65¢ in total, and we know she has one quarter (25¢). This means she needs to make up the remaining amount, which is:

65¢ - 25¢ = 40¢

Since she has no pennies, the remaining 40¢ must be made up of dimes (10¢) and nickels (5¢). Let's denote the number of dimes as (d) and the number of nickels as (n). The equation we need to satisfy is:

10d + 5n = 40

We can simplify this equation by dividing everything by 5:

2d + n = 8

Now, we can find the different combinations of dimes and nickels that satisfy this equation. We can express (n) in terms of (d):

n = 8 - 2d

Now, we need to find non-negative integer values for (d) and (n). The value of (d) can range from 0 to 4 (since (2d) must be less than or equal to 8):

1. If (d = 0):
   • (n = 8 - 2(0) = 8) → 0 dimes, 8 nickels
2. If (d = 1):
   • (n = 8 - 2(1) = 6) → 1 dime, 6 nickels
3. If (d = 2):
   • (n = 8 - 2(2) = 4) → 2 dimes, 4 nickels
4. If (d = 3):
   • (n = 8 - 2(3) = 2) → 3 dimes, 2 nickels
5. If (d = 4):
   • (n = 8 - 2(4) = 0) → 4 dimes, 0 nickels

Thus, the different coin combinations Tawny might have are:

1. 0 dimes and 8 nickels
2. 1 dime and 6 nickels
3. 2 dimes and 4 nickels
4. 3 dimes and 2 nickels
5. 4 dimes and 0 nickels

These combinations, along with the one quarter, give Tawny a total of 65¢.