How to make a dollar with nickels and dimes?

Steps to Solve

1. Identify the values of the coins

Nickels are worth $0.05$ and dimes are worth $0.10$.

2. Set up an equation

Let $n$ represent the number of nickels and $d$ represent the number of dimes. The total value of the coins can be expressed as: $$ 0.05n + 0.10d = 1.00 $$

3. Simplify the equation

To eliminate decimals, multiply the entire equation by $100$: $$ 5n + 10d = 100 $$

4. Rearrange the equation

You can simplify the equation by dividing everything by $5$: $$ n + 2d = 20 $$

5. Analyze possible solutions

Now, let's find combinations of $n$ (nickels) and $d$ (dimes):

• We can express $n$ in terms of $d$: $$ n = 20 - 2d $$

6. Determine valid values for dimes

For $n$ to be a non-negative number, $20 - 2d \geq 0$ which implies: $$ d \leq 10 $$

7. List the combinations

Now we can find the combinations:

• If $d = 0$, then $n = 20 - 2(0) = 20$
• If $d = 1$, then $n = 20 - 2(1) = 18$
• If $d = 2$, then $n = 20 - 2(2) = 16$
• Continuing this way up to $d = 10$ gives pairs of $(n, d)$: $$ (20, 0), (18, 1), (16, 2), (14, 3), (12, 4), (10, 5), (8, 6), (6, 7), (4, 8), (2, 9), (0, 10) $$