Mathematical Problem Solving: A Coin Riddle and Its Variants

Mathematics often involves solving word problems that require careful analysis and manipulation of equations. One classic example is the riddle about a piggy bank containing a specific number and combination of coins. Let's explore this problem and its variations, focusing on the application of algebraic equations to find solutions.

The Original Riddle

Consider the riddle where Patty has six times as many dimes as quarters in her piggy bank, and the total number of coins is 21, with a total value of $2.55. We can formulate the problem using algebra to find the number of each type of coin.

Formulating the Problem

Let q be the number of quarters Patty has. Since she has six times as many dimes as quarters, the number of dimes is 6q. The total number of coins can be expressed as:

q 6q 21

This simplifies to:

7q 21

Dividing both sides by 7 gives:

q 3

So, Patty has 3 quarters. The number of dimes is:

6q 6 * 3 18

Therefore, Patty has 18 dimes.

Calculating the Value of the Coins

Let's verify the total value of the coins:

The value of the quarters is:

3 * 0.25 0.75 dollars

The value of the dimes is:

18 * 0.10 1.80 dollars

Adding these amounts gives:

0.75 1.80 2.55 dollars

This matches the total amount given in the problem.

Exploring Variants of the Riddle

The original problem has a unique solution within the given constraints. However, it's worth exploring whether there are any other possible solutions that might exist:

1. Only Dimes and Quarters: The solution we found where Patty has 3 quarters and 18 dimes is the only valid solution under the given constraints. Any other combination would either exceed the total number of coins or the total value.

2. Including Other Coins: If we consider the possibility of including other types of coins, such as nickels or pennies, the problem becomes more complex. For instance, assuming Patty has 2 dollars, 9 nickels, and 10 cents, we can see that these also sum to 255 cents (2.55 dollars) and 21 coins.

Let's analyze this:

2 dollars 200 cents 9 nickels 45 cents 10 cents 10 cents 200 45 10 255 cents 21 coins in total

This shows that while the problem allows for creative solutions, the original constraints focusing on quarters and dimes lead to a unique and specific solution.

Conclusion

Math problems like this one involve logical reasoning and the application of algebraic equations. While the original riddle has a unique solution under the given constraints, considering variations can lead to a deeper understanding of problem-solving techniques and the importance of carefully defined conditions.