There are some students in an auditorium, some of them are dressed in white and the others are drcssed in bluc. The ratio of the number of boys dressed in white to the number of girls dressed in blue is `4 : 3` and the ratio of the total number of girls dressed in white to the number of boys dressed in blue is ` 4 : 5`. The ratio of the total number of boys and girls dressed in white to the total number of boys and girls dressed in blue is ` 12 : 13`. Find the ratio of the total number of boys to that of girls in the auditorium.

Options

A. `13 : 15`
B. ` 19 : 17`
C. ` 14 : 11`
D. Cannot be determined

Explanation

To solve the problem step by step, we can denote the variables based on the given ratios:

1. **Define Variables:**
- Let the number of boys dressed in white be \( 4x \).
- Let the number of girls dressed in blue be \( 3x \).
- Let the number of girls dressed in white be \( 4y \).
- Let the number of boys dressed in blue be \( 5y \).

2. **Write the Ratios:**
- From the problem, we have:
- Boys in white to girls in blue: \( \frac{4x}{3x} \) (which simplifies to \( 4:3 \)).
- Girls in white to boys in blue: \( \frac{4y}{5y} \) (which simplifies to \( 4:5 \)).
- Total boys and girls in white to total boys and girls in blue: \( \frac{4x + 4y}{5y + 3x} = \frac{12}{13} \).

3. **Set Up the Equation:**
- From the last ratio, we can set up the equation:
\[ \frac{4x + 4y}{5y + 3x} = \frac{12}{13} \]
- Cross-multiplying gives:
\[ 13(4x + 4y) = 12(5y + 3x) \]

4. **Expand and Simplify:**
- Expanding both sides:
\[ 52x + 52y = 60y + 36x \]
- Rearranging terms:
\[ 52x - 36x = 60y - 52y \]
\[ 16x = 8y \]
- Dividing both sides by 8:
\[ 2x = y \quad \text{or} \quad \frac{x}{y} = \frac{1}{2} \]

5. **Calculate Total Boys and Girls:**
- Total boys (white and blue):
\[ 4x + 5y \]
- Total girls (white and blue):
\[ 4y + 3x \]

6. **Substituting \( y \) in Terms of \( x \):**
- Since \( y = 2x \):
- Total boys:
\[ 4x + 5(2x) = 4x + 10x = 14x \]
- Total girls:
\[ 4(2x) + 3x = 8x + 3x = 11x \]

7. **Finding the Ratio of Boys to Girls:**
- The ratio of total boys to total girls:
\[ \frac{14x}{11x} = \frac{14}{11} \]

Thus, the ratio of the total number of boys to that of girls in the auditorium is **14:11**.