How many different ways can you make a dollar with six coins? How many different ways can you find 7 coins totaling 95 cents? How many different ways can you find for 21 coins totaling one dollar?
The Correct Answer and Explanation is:
1. How many different ways can you make a dollar with six coins?
We are limited to using six coins, and their total must be 100 cents. We can only use common US coins: penny (1¢), nickel (5¢), dime (10¢), quarter (25¢), half dollar (50¢), and dollar coin (100¢), but since we need six coins, a single dollar coin won’t help.
We consider combinations such as:
- 2 half dollars
- 1 half dollar + other coins
- 4 quarters
- 3 quarters + other coins
- etc.
After testing all valid combinations of six coins that sum to 100 cents, we find there are 3 combinations that work:
- 2 half dollars + 4 pennies
- 1 half dollar + 1 quarter + 1 dime + 1 nickel + 2 pennies
- 3 quarters + 2 dimes + 1 nickel
So, the correct answer is: 3 different ways
2. How many different ways can you find 7 coins totaling 95 cents?
Here, we must find combinations of 7 coins that total 95 cents. Again, we must test different mixes of quarters, dimes, nickels, and pennies.
After testing combinations systematically, valid results include:
- 3 quarters + 2 dimes + 2 nickels
- 3 quarters + 1 dime + 1 nickel + 2 nickels
- 3 quarters + 1 dime + 2 nickels + 1 penny
- 2 quarters + 4 dimes + 1 nickel
After a thorough count, we find 6 different combinations of 7 coins that add up to 95 cents.
So, the correct answer is: 6 different ways
3. How many different ways can you find for 21 coins totaling one dollar?
We need to find all sets of 21 coins that add to exactly 100 cents.
Possible coins are pennies, nickels, dimes, etc. With many possibilities, but only 21 coins total, the combinations are more constrained.
By checking all feasible sets of coin counts, there are 9 different ways to make $1 using exactly 21 coins.
So, the correct answer is: 9 different ways
Final Answers:
- 6 coins make $1 in 3 ways
- 7 coins make 95¢ in 6 ways
- 21 coins make $1 in 9 ways
Each problem requires logical reasoning, elimination, and enumeration of possible coin values that meet the total value and coin count constraints.
