Question

In how many different ways can you make change for 55 cents using nothing but nickel and / or dimes

Answer 1

To find the different ways to make change for 55 cents using only nickels (5 cents) and dimes (10 cents), we can set up an equation.

Let \( n \) be the number of nickels and \( d \) be the number of dimes. The value of the change can be expressed as:

\[
5n + 10d = 55
\]

We can simplify this equation by dividing everything by 5:

\[
n + 2d = 11
\]

Now, we will express \( n \) in terms of \( d \):

\[
n = 11 - 2d
\]

To ensure \( n \) is a non-negative integer, the right side of the equation \( 11 - 2d \) must be non-negative:

\[
11 - 2d \geq 0
\]

This simplifies to:

\[
11 \geq 2d \quad \text{or} \quad d \leq 5.5
\]

Since \( d \) must be a whole number, the maximum possible value for \( d \) is 5. Therefore, \( d \) can take any integer value from 0 to 5.

Now, let's enumerate the possible values of \( d \) and the corresponding \( n \):

- If \( d = 0 \), then \( n = 11 - 2(0) = 11 \).
- If \( d = 1 \), then \( n = 11 - 2(1) = 9 \).
- If \( d = 2 \), then \( n = 11 - 2(2) = 7 \).
- If \( d = 3 \), then \( n = 11 - 2(3) = 5 \).
- If \( d = 4 \), then \( n = 11 - 2(4) = 3 \).
- If \( d = 5 \), then \( n = 11 - 2(5) = 1 \).

Thus, the pairs \((n, d)\) that satisfy the equation are:

1. \( (11, 0) \)
2. \( (9, 1) \)
3. \( (7, 2) \)
4. \( (5, 3) \)
5. \( (3, 4) \)
6. \( (1, 5) \)

This gives us a total of 6 different combinations of nickels and dimes that can make change for 55 cents.

Thus, the final answer is:

\[
\boxed{6}
\]