Tickets for a concert cost `$4.00` for children and `$6.00` for adults. 850 concert tickets were sold for a total cost of `$3820`. How many children's tickets were sold?

Answer

To solve the problem, we will set up a system of equations based on the information given about the concert tickets.

### Step 1: Define Variables
Let:
- \( x \) = number of children's tickets sold
- \( y \) = number of adult tickets sold

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of tickets sold (children's + adults) is 850:
\[ x + y = 850 \quad \text{(Equation 1)} \]

2. The total revenue from the tickets sold is $3820. Since children's tickets cost $4 and adult tickets cost $6, we can express this as:
\[ 4x + 6y = 3820 \quad \text{(Equation 2)} \]

### Step 3: Solve Equation 1 for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 850 - x \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 2
Now, substitute Equation 3 into Equation 2:
\[ 4x + 6(850 - x) = 3820 \]
Expanding this gives:
\[ 4x + 5100 - 6x = 3820 \]

### Step 5: Combine Like Terms
Combine the \( x \) terms:
\[ -2x + 5100 = 3820 \]

### Step 6: Isolate \( x \)
Subtract 5100 from both sides:
\[ -2x = 3820 - 5100 \]
\[ -2x = -1280 \]
Now, divide by -2:
\[ x = \frac{1280}{2} = 640 \]

### Step 7: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 3:
\[ y = 850 - 640 = 210 \]

### Conclusion
The number of children's tickets sold is \( \boxed{640} \).