There are some boys and girls in a room. The square of the number of 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 boys and girls in the room is

## Concept & Logic This problem relies on translating verbal statements into basic algebraic equations and utilizing the difference of squares identity. The key formula required is: $a^2 - b^2 = (a - b)(a + b)$ ## Step-by-Step Solution * Let the number of boys be $b$ and the number of girls be $g$. * We are given that the square of the number of girls is less than the square of the number of boys by 28. Equation 1: $b^2 - g^2 = 28$ * We are also given that if there were two more girls, the number of boys would equal the number of girls. Equation 2: $b = g + 2$ Rearranging this gives: $b - g = 2$ * Expand Equation 1 using the difference of squares formula: $(b - g)(b + g) = 28$ * Substitute the value of $(b - g)$ from Equation 2 into the expanded equation: $2(b + g) = 28$ * Divide by 2 to isolate the sum: $b + g = 14$ * The expression $(b + g)$ represents exactly the total number of boys and girls in the room. ## Exam Strategy & Shortcut Instead of solving for individual values of $b$ and $g$, notice that the question asks for the **total number** of students, which is $(b + g)$. By dividing the difference of squares $(b^2 - g^2 = 28)$ by the simple difference $(b - g = 2)$, you immediately find the sum $(b + g = 14)$ in under 10 seconds. ## Common Pitfall A major mistake is trying to solve for $b$ and $g$ individually (e.g., finding $b=8, g=6$), which wastes precious time. Always keep your eye on what the question is specifically asking for before proceeding with full variable isolation. ## Final Answer **Therefore, the correct answer is 14.**