Solved problems: Choosing coins to make under $1

🪙 Understanding the Coin Problem

The coin problem, in its simplest form, involves determining the different combinations of coins (pennies, nickels, dimes, quarters) that can sum up to a specific target amount, usually less than a dollar. This problem explores concepts of number theory and combinatorics, providing a practical application of mathematical principles.

📜 History and Background

Coin problems have been around for centuries, often appearing as recreational math puzzles. They touch upon the fundamental aspects of representing numbers in different bases, similar to how computers use binary code. While not tied to a specific historical figure, the problem embodies the spirit of mathematical exploration and problem-solving that has driven mathematical inquiry for ages.

🔑 Key Principles

• 💰 Greedy Algorithm Limitation: The 'greedy' approach (using the largest denomination coin first) doesn't always yield the optimal (fewest coins) or all possible solutions.
• 🧮 Systematic Approach: A structured method is crucial to avoid missing combinations. Consider starting with the largest coin denomination and working downwards.
• ➕ Equation Representation: The problem can be represented as a linear equation: $p + 5n + 10d + 25q = amount$, where $p$ = pennies, $n$ = nickels, $d$ = dimes, and $q$ = quarters.
• 🔢 Integer Solutions: We are only interested in non-negative integer solutions since we can't have fractions of coins.

✍️ Solving Coin Combination Problems

Here's a step-by-step approach to finding all possible combinations of coins to reach a specific amount (under $1):

1. 🎯 Define the Target: Clearly state the amount you want to achieve (e.g., 47 cents).
2. 🧱 Start with Quarters: Determine the maximum number of quarters that can be used without exceeding the target amount.
3. ➗ Iterate Through Possibilities: For each possible number of quarters, calculate the remaining amount. Then, find the possible number of dimes for that remaining amount, and so on.
4. 📝 Record Combinations: Systematically list each valid combination of (quarters, dimes, nickels, pennies).

🧮 Example: Making 27 Cents

Let's find all the ways to make 27 cents using pennies, nickels, dimes, and quarters.

1. Quarters: Maximum 1 quarter (25 cents).
2. Case 1: 1 Quarter
   • Remaining: 2 cents.
   • Dimes: 0
   • Nickels: 0
   • Pennies: 2
   • Combination: (1 Quarter, 0 Dimes, 0 Nickels, 2 Pennies)
3. Case 2: 0 Quarters
   • Remaining: 27 cents.
   • Dimes: Max 2 dimes (20 cents)
   • Subcase 1: 2 Dimes
     • Remaining: 7 cents.
     • Nickels: 1
     • Pennies: 2
     • Combination: (0 Quarters, 2 Dimes, 1 Nickel, 2 Pennies)
   • Subcase 2: 2 Dimes
     • Remaining: 7 cents.
     • Nickels: 0
     • Pennies: 7
     • Combination: (0 Quarters, 2 Dimes, 0 Nickel, 7 Pennies)
   • Subcase 3: 1 Dime
     • Remaining: 17 cents.
     • Nickels: 3
     • Pennies: 2
     • Combination: (0 Quarters, 1 Dime, 3 Nickels, 2 Pennies)
   • Subcase 4: 1 Dime
     • Remaining: 17 cents.
     • Nickels: 2
     • Pennies: 7
     • Combination: (0 Quarters, 1 Dime, 2 Nickels, 7 Pennies)
   • Subcase 5: 1 Dime
     • Remaining: 17 cents.
     • Nickels: 1
     • Pennies: 12
     • Combination: (0 Quarters, 1 Dime, 1 Nickels, 12 Pennies)
   • Subcase 6: 1 Dime
     • Remaining: 17 cents.
     • Nickels: 0
     • Pennies: 17
     • Combination: (0 Quarters, 1 Dime, 0 Nickels, 17 Pennies)
   • Subcase 7: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 5
     • Pennies: 2
     • Combination: (0 Quarters, 0 Dime, 5 Nickels, 2 Pennies)
   • Subcase 8: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 4
     • Pennies: 7
     • Combination: (0 Quarters, 0 Dime, 4 Nickels, 7 Pennies)
   • Subcase 9: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 3
     • Pennies: 12
     • Combination: (0 Quarters, 0 Dime, 3 Nickels, 12 Pennies)
   • Subcase 10: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 2
     • Pennies: 17
     • Combination: (0 Quarters, 0 Dime, 2 Nickels, 17 Pennies)
   • Subcase 11: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 1
     • Pennies: 22
     • Combination: (0 Quarters, 0 Dime, 1 Nickels, 22 Pennies)
   • Subcase 12: 0 Dime
     • Remaining: 27 cents.
     • Nickels: 0
     • Pennies: 27
     • Combination: (0 Quarters, 0 Dime, 0 Nickels, 27 Pennies)

💡 Tips and Tricks

• 🗺️ Organization is Key: Using a table or chart to organize your combinations can help avoid errors.
• 🔎 Double-Check: Always verify that the sum of your chosen coins equals the target amount.
• 🧠 Look for Patterns: As you solve more problems, you may notice patterns that can simplify the process.

📝 Practice Quiz

Find all the possible combinations of coins to make the following amounts:

1. 1. 17 cents
2. 2. 33 cents
3. 3. 52 cents
4. 4. 68 cents
5. 5. 81 cents
6. 6. 94 cents
7. 7. 99 cents

🌍 Real-World Applications

• 🛒 Making Change: Understanding coin combinations is essential for cashiers and anyone handling money.
• 🕹️ Vending Machines: Vending machines use algorithms to determine the correct change to dispense based on the coins inserted.
• 🏦 Financial Literacy: This problem helps develop skills in financial literacy and budgeting.

🎓 Conclusion

The coin problem is a versatile and engaging mathematical challenge that combines number theory, problem-solving, and practical application. By understanding the key principles and adopting a systematic approach, you can master this problem and enhance your mathematical skills. Happy counting!