A group of 15 friends went out for dinner. The cost per meal was $12 for children and $20 for adults. The total bill came to $252. How many adults were in the group?

Let's denote:

- $ x $ as the number of children in the group.
- $ y $ as the number of adults in the group.

We are given the following information:

1. The total number of people is 15:
   \[
   x + y = 15
   \]
2. The total cost of the meals is $252, with each child's meal costing $12 and each adult's meal costing $20:
   \[
   12x + 20y = 252
   \]

### Step 1: Solve the system of equations
We have two equations:

\[
\begin{aligned}
1) & \quad x + y = 15 \\
2) & \quad 12x + 20y = 252
\end{aligned}
\]

### Step 2: Solve the first equation for $ y $:
\[
y = 15 - x
\]

### Step 3: Substitute $ y = 15 - x $ into the second equation:
\[
12x + 20(15 - x) = 252
\]

### Step 4: Distribute and simplify:
\[
12x + 300 - 20x = 252
\]

Combine like terms:
\[
-8x + 300 = 252
\]

### Step 5: Solve for $ x $:
\[
-8x = 252 - 300
\]
\[
-8x = -48
\]
\[
x = \frac{-48}{-8} = 6
\]

So, there are 6 children in the group.

### Step 6: Find the number of adults:
\[
y = 15 - x = 15 - 6 = 9
\]

Thus, there are **9 adults** in the group.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. If the total number of friends was different, how would that affect the number of adults?
2. How would the problem change if the meal costs were different?
3. What if the total bill was higher? How would you solve for the number of adults then?
4. How could you express this problem using a different method, such as matrices?
5. What if there was an additional discount on the total bill? How would that impact the solution?

**Tip:** When solving word problems involving systems of equations, clearly define your variables and equations before starting the calculations. This makes it easier to track and solve for unknowns.

Learn how to solve a word problem using systems of equations. Find out the number of adults in a group of 15 friends where meal costs vary between children and adults. Includes step-by-step solution and key algebraic concepts.

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