Question: Sarah had 27 nickels and quarters worth 4.75. How many of each type of coin did she have?
1. **State the problem:** Sarah has a total of $27$ coins consisting of nickels and quarters, and their total value is $4.75$. We need to find how many nickels and quarters she has.
2. **Define variables:** Let $n$ be the number of nickels and $q$ be the number of quarters.
3. **Write the equations:**
- Total coins: $$n + q = 27$$
- Total value in cents (since $4.75 = 475$ cents): $$5n + 25q = 475$$
4. **Solve the system:**
From the first equation, express $n$ as:
$$n = 27 - q$$
Substitute into the second equation:
$$5(27 - q) + 25q = 475$$
5. **Simplify and solve for $q$:**
$$135 - 5q + 25q = 475$$
$$135 + 20q = 475$$
$$20q = 475 - 135$$
$$20q = 340$$
$$q = \frac{\cancel{20}q}{\cancel{20}} = \frac{340}{20} = 17$$
6. **Find $n$:**
$$n = 27 - 17 = 10$$
7. **Answer:** Sarah has 10 nickels and 17 quarters.
**Final answer:** 10,17