In the `10^(th)` grade of Brooklyn Public School, the ratio of the number of boys to the number of girls was 3: 5 . Among the students , some had taken up literature as a specialzation while the rest had taken up science. The ratio of the number of literature students to science students was 5: 7 . If it is known that one - thrid the number of boys had taken up science as a specialization , what fraction of the girls had literature as their specialization ?

Options

A. `(4)/(15)`
B. `(2)/(5)`
C. `(3)/(5)`
D. `(2)/(3)`

Answer

To solve the problem step by step, we will define variables and use the given ratios to find the required fraction of girls who had literature as their specialization.

### Step 1: Define Variables
Let:
- \( B_L \) = number of boys taking literature
- \( B_S \) = number of boys taking science
- \( G_L \) = number of girls taking literature
- \( G_S \) = number of girls taking science

### Step 2: Set Up the Ratios
From the problem, we know:
1. The ratio of boys to girls is \( 3:5 \).
\[ \frac{B_L + B_S}{G_L + G_S} = \frac{3}{5} \]
We can express this as:
\[ B_L + B_S = 3k \quad \text{and} \quad G_L + G_S = 5k \quad \text{for some } k \]

2. The ratio of literature students to science students is \( 5:7 \).
\[ \frac{B_L + G_L}{B_S + G_S} = \frac{5}{7} \]
This can be expressed as:
\[ B_L + G_L = 5m \quad \text{and} \quad B_S + G_S = 7m \quad \text{for some } m \]

3. One-third of the boys took science.
\[ B_S = \frac{1}{3}(B_L + B_S) \]
This implies:
\[ B_S = x \quad \text{and} \quad B_L + B_S = 3x \quad \Rightarrow \quad B_L = 2x \]

### Step 3: Substitute Values
From the equations we have:
- \( B_L = 2x \)
- \( B_S = x \)

Now we can substitute these values into the first ratio:
\[ B_L + B_S = 2x + x = 3x \quad \text{and} \quad G_L + G_S = 5k \]

### Step 4: Use the First Ratio
Substituting into the first equation:
\[ \frac{3x}{G_L + G_S} = \frac{3}{5} \]
Cross-multiplying gives:
\[ 5(3x) = 3(G_L + G_S) \quad \Rightarrow \quad G_L + G_S = 5x \]

### Step 5: Use the Second Ratio
Now substituting into the second ratio:
\[ B_L + G_L = 2x + G_L \quad \text{and} \quad B_S + G_S = x + G_S \]
Thus:
\[ \frac{2x + G_L}{x + G_S} = \frac{5}{7} \]
Cross-multiplying gives:
\[ 7(2x + G_L) = 5(x + G_S) \]
Expanding this:
\[ 14x + 7G_L = 5x + 5G_S \]
Rearranging gives:
\[ 9x = 5G_S - 7G_L \]

### Step 6: Substitute for \( G_S \)
From \( G_L + G_S = 5x \), we can express \( G_S \) as:
\[ G_S = 5x - G_L \]
Substituting \( G_S \) into the equation:
\[ 9x = 5(5x - G_L) - 7G_L \]
This simplifies to:
\[ 9x = 25x - 5G_L - 7G_L \quad \Rightarrow \quad 9x = 25x - 12G_L \]
Rearranging gives:
\[ 12G_L = 16x \quad \Rightarrow \quad G_L = \frac{4}{3}x \]

### Step 7: Find the Total Number of Girls
Using \( G_L + G_S = 5x \):
\[ G_S = 5x - G_L = 5x - \frac{4}{3}x = \frac{15x - 4x}{3} = \frac{11}{3}x \]

### Step 8: Calculate the Fraction of Girls in Literature
The fraction of girls who took literature is:
\[ \frac{G_L}{G_L + G_S} = \frac{\frac{4}{3}x}{\frac{4}{3}x + \frac{11}{3}x} = \frac{\frac{4}{3}x}{\frac{15}{3}x} = \frac{4}{15} \]

### Final Answer
Thus, the fraction of girls who had literature as their specialization is:
\[ \frac{4}{15} \]