At a certain mive theate, there are 16 rows and each row has either 20 or 24 seats.If the total number of seats in all 16 rows is 348, how many rows have 24 seats ?

Options

A. 7
B. 9
C. 11
D. 13

Explanation

To solve the problem step by step, we will use algebraic equations based on the information provided.

### Step 1: Define Variables
Let:
- \( X \) = number of rows with 20 seats
- \( Y \) = number of rows with 24 seats

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of rows is 16:
\[ X + Y = 16 \quad \text{(Equation 1)} \]

2. The total number of seats is 348:
\[ 20X + 24Y = 348 \quad \text{(Equation 2)} \]

### Step 3: Simplify the Second Equation
To make calculations easier, we can divide the entire second equation by 4:
\[ 5X + 6Y = 87 \quad \text{(Equation 3)} \]

### Step 4: Solve for One Variable
From Equation 1, we can express \( X \) in terms of \( Y \):
\[ X = 16 - Y \]

### Step 5: Substitute into the Third Equation
Now, substitute \( X \) in Equation 3:
\[ 5(16 - Y) + 6Y = 87 \]
Expanding this gives:
\[ 80 - 5Y + 6Y = 87 \]

### Step 6: Combine Like Terms
Combine the \( Y \) terms:
\[ 80 + Y = 87 \]

### Step 7: Solve for \( Y \)
Now, isolate \( Y \):
\[ Y = 87 - 80 \]
\[ Y = 7 \]

### Step 8: Find \( X \)
Now that we have \( Y \), we can find \( X \):
\[ X = 16 - Y = 16 - 7 = 9 \]

### Conclusion
Thus, the number of rows with 24 seats is \( Y = 7 \).