https://file.aiquickdraw.com/mathbot/file/1727797378385v3nv4x43.jpg

Let's break down the problem into a system of equations:

### Given Information:
- A cup of ice cream costs $2.00.
- A pint of ice cream costs $3.50.
- 30 customers each buy either a cup or a pint, so the total number of items sold is 30.
- The total sales for the hour is $84.

### Define Variables:
- Let $ x $ represent the number of cups sold.
- Let $ y $ represent the number of pints sold.

### Set up the system of equations:
1. Since 30 customers bought either a cup or a pint:
   \[
   x + y = 30
   \]
2. The total sales come from the cups and pints sold, so:
   \[
   2x + 3.5y = 84
   \]

### Solve the System of Equations:
We can solve this system by substitution or elimination. I'll use substitution.

From the first equation:
\[
x = 30 - y
\]
Substitute $ x = 30 - y $ into the second equation:
\[
2(30 - y) + 3.5y = 84
\]
Distribute the 2:
\[
60 - 2y + 3.5y = 84
\]
Combine like terms:
\[
60 + 1.5y = 84
\]
Subtract 60 from both sides:
\[
1.5y = 24
\]
Solve for $ y $:
\[
y = \frac{24}{1.5} = 16
\]

Now substitute $ y = 16 $ back into the first equation:
\[
x + 16 = 30 \quad \Rightarrow \quad x = 14
\]

### Solution:
- 14 cups and 16 pints of ice cream were sold.

The correct answer is: **14 cups and 16 pints**.

---

Do you have any questions or would you like further details? Here are five follow-up questions:

1. How would you solve this using the elimination method?
2. Can you create a similar problem with different prices and solve it?
3. What would happen if the total sales were higher, say $100, but the same number of items sold?
4. How can we graph the system of equations and interpret the solution?
5. What is the effect of changing the price of the pint to $4 instead of $3.50?

**Tip:** When solving a system of equations, substitution is often useful when one equation is already solved for one variable.

An ice cream shop sells ice cream by cups and pints. Each cup of ice cream costs $2.00, and each pint of ice cream costs $3.50. Over the course of one hour, 30 customers each buy one item, and the total sales for the hour are $84. How many cups and pints of ice cream are sold during the one-hour period?

Learn how to solve a system of equations for an ice cream shop's sales problem. Given prices of cups and pints of ice cream and total sales, find how many cups and pints were sold. This problem covers key algebraic concepts, including substitution and solving linear systems. Suitable for students in Grades 7-10.

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