In Miss Hoover's class, the ratio of boys to girls is x to y. If the total number of children in the class is five times the number of boys in the class, and x and y are integers, which of the following could be the sum of x and y?

Options

A. 9
B. 10
C. 11
D. 12

Answer

To solve the problem step by step, we will analyze the given information and derive the necessary equations.

### Step 1: Define Variables
Let:
- \( x \) = number of boys
- \( y \) = number of girls

The ratio of boys to girls is given as \( x : y \).

### Step 2: Total Number of Children
The total number of children in the class can be expressed as:
\[ \text{Total children} = x + y \]

### Step 3: Relationship Between Total Children and Boys
According to the problem, the total number of children is five times the number of boys:
\[ x + y = 5x \]

### Step 4: Rearranging the Equation
Now, we can rearrange the equation to isolate \( y \):
\[ y = 5x - x \]
\[ y = 4x \]

### Step 5: Sum of x and y
Now we can find the sum \( x + y \):
\[ x + y = x + 4x = 5x \]

### Step 6: Divisibility Condition
Since \( x + y = 5x \), it is clear that \( x + y \) must be divisible by 5. Therefore, \( x + y \) is a multiple of 5.

### Step 7: Possible Values for x + y
Now we need to determine which of the provided options could be the sum of \( x \) and \( y \). Since \( x + y \) must be divisible by 5, we can only consider those options that meet this criterion.

### Conclusion
The only possible value for \( x + y \) that is divisible by 5 among the given options is 10.

### Final Answer
Thus, the sum of \( x \) and \( y \) could be:
\[ \boxed{10} \]