There are some boys and girls in a room. The square of the number of the girls is less than the square of the number of boys by 28. If there were two more girls, the number of boys would have been the same as that of the girls. The total number of the boys and girls in the room are

Options

A. 56
B. 14
C. 10
D. 7

Answer

To solve the problem step by step, we will define variables for the number of boys and girls and set up equations based on the information provided.

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

### Step 2: Set Up the First Equation
According to the problem, the square of the number of girls is less than the square of the number of boys by 28. This can be expressed mathematically as:
\[ X^2 - Y^2 = 28 \]

### Step 3: Set Up the Second Equation
The problem also states that if there were two more girls, the number of boys would be equal to the number of girls. This can be expressed as:
\[ X = Y + 2 \]
Alternatively, we can rearrange this to:
\[ X - Y = 2 \]

### Step 4: Substitute the Second Equation into the First
We can use the second equation \( X - Y = 2 \) to express \( X \) in terms of \( Y \):
\[ X = Y + 2 \]

Now, substitute \( X \) into the first equation:
\[ (Y + 2)^2 - Y^2 = 28 \]

### Step 5: Expand and Simplify
Expanding the left side:
\[ (Y^2 + 4Y + 4) - Y^2 = 28 \]
This simplifies to:
\[ 4Y + 4 = 28 \]

### Step 6: Solve for \( Y \)
Subtract 4 from both sides:
\[ 4Y = 24 \]
Now, divide by 4:
\[ Y = 6 \]

### Step 7: Find \( X \)
Now that we have \( Y \), we can find \( X \) using the equation \( X = Y + 2 \):
\[ X = 6 + 2 = 8 \]

### Step 8: Calculate the Total Number of Boys and Girls
Now, we can find the total number of boys and girls:
\[ X + Y = 8 + 6 = 14 \]

### Final Answer
The total number of boys and girls in the room is:
\[ \boxed{14} \]