Question

How many ways are there to make $2.00 dollars from only nickels and quarters? (Hints: converts this to a linear Diophantine equation. By its nature, solutions will be non-negative).

Answer 1

There are 9 ways to make $2.00 with nickels and quarters.

Step-by-step explanation:

To solve this problem, we can set up a linear Diophantine equation, which is an equation where we are looking for integer solutions. Let
`\(n\)`represent the number of nickels and
`\(q\)` represent the number of quarters. The value of nickels is 5 cents, and quarters are 25 cents, so the equation becomes
`\(5n + 25q = 200\)`. Dividing both sides by 5 gives us
`\(n + 5q = 40\)`. This equation represents the possible combinations of nickels and quarters that sum up to $2.00.

Now, we can find the solutions for
`\(n\)` and
`\(q\)`. Notice that
`\(n\)`must be a multiple of 5, as it represents nickels. We can start with
`\(n = 0\)`and find the corresponding values of
`\(q\)`. The valid solutions are
`\(n = 0, 5, 10, 15, 20, 25, 30, 35, 40\)`, which gives us 9 possible ways to make $2.00 using only nickels and quarters.

In summary, by converting the problem into a linear Diophantine equation and considering the nature of the solutions to be non-negative, we find that there are 9 ways to make $2.00 with nickels and quarters.