To find out how many ways we can make 1 dollar using just nickels (5 cents) and dimes (10 cents), we can approach the problem by considering the amount of each coin used.
Let’s denote:
- n = number of nickels
- d = number of dimes
We want the total value to equal 100 cents (1 dollar). Therefore, we have the equation:
5n + 10d = 100
We can simplify this equation by dividing everything by 5:
n + 2d = 20
This means for any number of dimes (‘d’), we can find the corresponding number of nickels (‘n’). We can express ‘n’ in terms of ‘d’:
n = 20 – 2d
Now, since ‘n’ must be a non-negative integer, we need:
20 – 2d \\geq 0
Which leads to:
d \\leq 10
This shows that ‘d’ can take on values from 0 to 10 (inclusive). So, let’s count the possibilities:
- For d = 0: n = 20 (0 dimes, 20 nickels)
- For d = 1: n = 18 (1 dime, 18 nickels)
- For d = 2: n = 16 (2 dimes, 16 nickels)
- For d = 3: n = 14 (3 dimes, 14 nickels)
- For d = 4: n = 12 (4 dimes, 12 nickels)
- For d = 5: n = 10 (5 dimes, 10 nickels)
- For d = 6: n = 8 (6 dimes, 8 nickels)
- For d = 7: n = 6 (7 dimes, 6 nickels)
- For d = 8: n = 4 (8 dimes, 4 nickels)
- For d = 9: n = 2 (9 dimes, 2 nickels)
- For d = 10: n = 0 (10 dimes, 0 nickels)
Thus, for every integer value of ‘d’ from 0 to 10, there is a corresponding value of ‘n’. Since ‘d’ can take on 11 different values (0 through 10), there are a total of 11 ways to make a dollar using just nickels and dimes.
In summary:
The total number of ways to make 1 dollar using just nickels and dimes is 11.