Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Ruth and Heather met for dinner in downtown Bluepoint. Ruth did flat-rate valet parking for $10. Heather went to another valet and paid $8 up front and $1 for every hour, including the first hour. Ultimately, the friends ended up paying the same amount. How long did they stay? How much did each one pay?


For what hours of parking, Ruth and Heather both ended up paying how much money
.

Answer 1

To solve this problem, we will set up a system of equations based on the information provided about the parking costs for Ruth and Heather.

Let \( h \) represent the number of hours they parked.

**Ruth's parking cost:**
- Ruth pays a flat rate of $10, so her total cost is:
\[ R = 10 \]

**Heather's parking cost:**
- Heather pays $8 upfront plus $1 for every hour she parks, which gives her total cost as:
\[ H = 8 + 1h \]

According to the problem, Ruth and Heather ended up paying the same amount. Thus, we can set the equations equal to each other:
\[
10 = 8 + 1h
\]

Now we will solve for \( h \).

1. Subtract 8 from both sides:
\[
10 - 8 = 1h
\]
\[
2 = 1h
\]
\[
h = 2
\]

Now that we have found the number of hours they parked (\( h \)), we can calculate how much each one paid:

- **Ruth paid:**
\[
R = 10
\]

- **Heather paid:**
\[
H = 8 + 1h = 8 + 1(2) = 8 + 2 = 10
\]

So, both Ruth and Heather paid $10.

**To fill in the blanks:**
- For what hours of parking, Ruth and Heather both ended up paying how much money:
- They stayed for **2 hours**.
- Each one paid **$10**.