Coin Exchange Puzzle: Dimes and Quarters

Have you ever come across a puzzle that left you scratching your head? Today, let's dive into one such riddle: Someone exchanged a dollar bill for change and received 7 coins, none of which were half-dollars. How many of these coins were dimes? The choices are:

a. 0 b. 1 c. 4 d. 5 e. Cannot be determined from the information given

At first glance, it seems straightforward. However, the solution is not as simple as it may appear. Let's break it down step-by-step to understand why the answer is 'e. Cannot be determined from the information given.'

Understanding the Problem

The problem states that the person received 7 coins, all of which were either quarters, nickels, or dimes, but not half-dollars. They received a total of $1.00 in change. Let's explore the possible combinations of coins that could make up this change.

Option A: 5 Dimes

Let's first consider if it's possible to have 5 dimes:

5 dimes 50 cents

Since the total amount of change is $1.00, the remaining amount is:

$1.00 - $0.50 $0.50, or 50 cents.

Now, the problem specifies that there are 7 coins in total. If we have 5 dimes, we need 2 more coins to make up the total of 7 coins. The remaining 50 cents can be made up with:

2 quarters 50 cents, but this is not possible since none of the coins should be quarters. 1 nickel and 1 quarter, but again, this is not possible. 10 nickels, which is not possible since the total would be too high (50 10 x 5 100 cents). 10 nickels, but this is not possible since the total would be too high (50 10 x 5 100 cents).

Therefore, it's not possible to have 5 dimes in this scenario.

Option B: 3 Quarters and 2 Dimes

Let's consider another possibility with 3 quarters and 2 dimes:

3 quarters 75 cents 2 dimes 20 cents

The total amount is:

$0.75 $0.20 $0.95, which is not $1.00.

To make it $1.00, we need an additional 5 cents. This can be made up with one nickel, which brings the total to:

3 quarters 2 dimes 1 nickel 7 coins.

However, the problem specifies that none of the coins can be half-dollars, and this combination does not violate this condition.

Conclusion

From the above analysis, it is clear that the only possible combination that fits the criteria is:

3 quarters 1 dime 3 nickels 7 coins, making a total of $1.00.

However, another possible combination is:

5 dimes 2 nickels 7 coins, making a total of $1.00, but this does not fit the condition of one of the coins being a dime.

Therefore, the number of dimes cannot be determined with certainty from the given information. This is why the correct answer is:

Cannot be determined from the information given.

Additional Information

When solving such puzzles, it's essential to consider all possible combinations of coins and verify if they meet the criteria given. In this case, the ambiguity arises from the fact that there can be multiple valid combinations that satisfy the given conditions, but the exact number of dimes can vary.

This type of puzzle is not only a fun brain teaser but also an excellent way to improve logical thinking and problem-solving skills. It can be applied in various real-life scenarios, such as budgeting, accounting, and even in competitive math challenges.

By understanding these concepts, you can enhance your ability to handle similar situations and make informed decisions. Whether you're dealing with dollar bills or other financial matters, having a clear grasp of coin exchanges is always beneficial.

Conclusion

In summary, the problem of determining how many of the 7 coins (not including half-dollars) were dimes, given a total of $1.00, is Cannot be determined from the information given.

Stay tuned for more such puzzles and riddles to sharpen your problem-solving skills.