If the ratio of boys to girls in a class is B and the ratio of girls to boys is G, then 3 (B + G) is :

Options

A. equal to 3
B. less than 3
C. more than 3
D. less than `(1)/(3)`

Answer

To solve the problem, we need to express the ratios of boys to girls and girls to boys mathematically, and then calculate the value of \(3(B + G)\).

### Step-by-Step Solution:

1. **Define the Variables:**
Let the number of boys be \(X\) and the number of girls be \(Y\).

2. **Express the Ratios:**
- The ratio of boys to girls is given as \(B = \frac{X}{Y}\).
- The ratio of girls to boys is given as \(G = \frac{Y}{X}\).

3. **Calculate \(B + G\):**
- We can express \(B + G\) as:
\[ B + G = \frac{X}{Y} + \frac{Y}{X} \]
- To combine these fractions, we need a common denominator:
\[ B + G = \frac{X^2 + Y^2}{XY} \]

4. **Calculate \(3(B + G)\):**
- Now, we can find \(3(B + G)\):
\[ 3(B + G) = 3 \left(\frac{X^2 + Y^2}{XY}\right) = \frac{3(X^2 + Y^2)}{XY} \]

5. **Analyze the Expression:**
- Since \(X\) and \(Y\) are positive integers, we can analyze the term \(\frac{X^2 + Y^2}{XY}\).
- By the Cauchy-Schwarz inequality, we know that:
\[ (X + Y)^2 \leq 2(X^2 + Y^2) \]
- This implies that:
\[ \frac{X^2 + Y^2}{XY} \geq 1 \]
- Therefore, \(3(B + G) = \frac{3(X^2 + Y^2)}{XY} > 3\).

### Final Result:
Thus, we conclude that \(3(B + G) > 3\).