Question

What are the two possible combinations of coins that a restaurant cashier can give as change for $1 using no more than four coins and no coin smaller than a nickel or bigger than a quarter?.

Answer 1

To determine the combinations of coins that a restaurant cashier can give as change for $1 using no more than four coins, with the restriction that no coin is smaller than a nickel (5 cents) or bigger than a quarter (25 cents), we need to analyze the possible combinations of nickels (5 cents), dimes (10 cents), and quarters (25 cents).

Here are the relevant combinations that total $1 (100 cents):

1. **Four Quarters:**
- \(4 \times 25 = 100\) cents
- Combination: (25, 25, 25, 25)

2. **One Quarter and Three Dimes:**
- \(1 \times 25 + 3 \times 10 = 25 + 30 = 55\) cents
- Not valid.

3. **Two Quarters and Two Dimes:**
- \(2 \times 25 + 2 \times 10 = 50 + 20 = 70\) cents
- Not valid.

4. **Three Quarters and One Nickel:**
- \(3 \times 25 + 1 \times 5 = 75 + 5 = 80\) cents
- Not valid.

5. **One Quarter, One Dime, and Two Nickels:**
- \(1 \times 25 + 1 \times 10 + 2 \times 5 = 25 + 10 + 10 = 45\) cents
- Not valid.

After analyzing various combinations, the valid solutions are:

- **One quarter (25 cents) and three dimes (30 cents):**
\(1 \times 25 + 3 \times 10 = 25 + 30 = 55\) cents
- Combination: (25, 10, 10, 10)

- **Five dimes (10 cents):**
\(10 \times 10 = 100\) cents
- Combination: (10, 10, 10, 10)

So, it turns out there are only two valid combinations of coins that total to one dollar:

1. **Four Quarters**: (25, 25, 25, 25)
2. **Five Dimes**: (10, 10, 10, 10)

If you want to restrict to only combinations that sum up to $1, there are no other practical combinations within the constraints provided.