https://file.aiquickdraw.com/mathbot/file/17241125274450hqvbu2i.jpg

To solve the problem provided in the image, let's define the variables and set up equations.

### Problem:
- Admission to the cinema is $3 per child and $7 per adult.
- A group of 12 people paid a total of $64 for admission.
- We need to determine the number of children in the group.

### Step 1: Define Variables
Let:
- $ x $ be the number of children.
- $ y $ be the number of adults.

### Step 2: Set Up Equations
From the information provided, we can create the following two equations:

1. The total number of people:
   \[
   x + y = 12
   \]

2. The total cost of admission:
   \[
   3x + 7y = 64
   \]

### Step 3: Solve the System of Equations

1. Solve the first equation for $ y $:
   \[
   y = 12 - x
   \]

2. Substitute this expression for $ y $ into the second equation:
   \[
   3x + 7(12 - x) = 64
   \]

3. Distribute and simplify:
   \[
   3x + 84 - 7x = 64
   \]

4. Combine like terms:
   \[
   -4x + 84 = 64
   \]

5. Isolate $ x $:
   \[
   -4x = 64 - 84
   \]
   \[
   -4x = -20
   \]
   \[
   x = 5
   \]

So, there are **5 children** in the group.

### Step 4: Verify the Solution
If there are 5 children, then the number of adults is:
\[
y = 12 - 5 = 7
\]

Check the total cost:
\[
3(5) + 7(7) = 15 + 49 = 64
\]
The calculations are correct.

### Final Answer:
There are **5 children** in the group.

Would you like more details, or do you have any questions? 

Here are 5 related questions you might find interesting:
1. What would the equations look like if the prices were changed to $4 for a child and $6 for an adult?
2. How many adults would there be if the group consisted only of adults?
3. What is the ratio of the number of children to the number of adults in this scenario?
4. If the total cost was $70, how many children and adults would be in the group?
5. How would the total cost change if there were 3 more children and 2 fewer adults?

**Tip:** When solving word problems, always start by defining variables and writing down what you know in the form of equations. This makes it easier to organize your work and find the solution.

Learn how to solve a problem involving admission costs for children and adults. This problem covers equations, including the total number of people and the total cost of admission. Suitable for students in Grades 7-9.

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