An a track meet,(2)/(5) of the first-place finishers attended southport hiegh school,and (1)/(2)of them were girls. If (2)/(9) of the first place finishers who did NOY attend southport high school were girls,What fraction pert of the totle number of first-place finishers were boys?

Options

A. `(1)/(9)`
B. `(2)/(15)`
C. `(3)/(5)`
D. `(2)/(3)`

Answer

To solve the problem step by step, we will define variables and set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( X \) = total number of first-place finishers
- \( G_S \) = number of girls who attended Southport High School
- \( B_S \) = number of boys who attended Southport High School
- \( G_{NS} \) = number of girls who did not attend Southport High School
- \( B_{NS} \) = number of boys who did not attend Southport High School

### Step 2: Set Up Equations Based on Given Information
1. According to the problem, \( \frac{2}{5} \) of the first-place finishers attended Southport High School. Therefore:
\[ G_S + B_S = \frac{2}{5} X \]

2. It is stated that half of those who attended Southport High School were girls. Hence:
\[ G_S = \frac{1}{2} (G_S + B_S) = \frac{1}{2} \left(\frac{2}{5} X\right) = \frac{1}{5} X \]
Since \( G_S + B_S = \frac{2}{5} X \) and \( G_S = \frac{1}{5} X \), we can find \( B_S \):
\[ B_S = \frac{2}{5} X - \frac{1}{5} X = \frac{1}{5} X \]

### Step 3: Calculate the Number of First Place Finishers Who Did Not Attend Southport High School
The number of first-place finishers who did not attend Southport High School is:
\[ G_{NS} + B_{NS} = X - (G_S + B_S) = X - \frac{2}{5} X = \frac{3}{5} X \]

### Step 4: Set Up the Equation for Girls Who Did Not Attend Southport High School
According to the problem, \( \frac{2}{9} \) of the first-place finishers who did not attend Southport High School were girls:
\[ G_{NS} = \frac{2}{9} \left(\frac{3}{5} X\right) = \frac{2 \times 3}{9 \times 5} X = \frac{6}{45} X = \frac{2}{15} X \]

### Step 5: Calculate the Number of Boys Who Did Not Attend Southport High School
Using the total number of first-place finishers who did not attend Southport High School:
\[ B_{NS} = \frac{3}{5} X - G_{NS} = \frac{3}{5} X - \frac{2}{15} X \]
To subtract these fractions, we need a common denominator:
\[ \frac{3}{5} X = \frac{9}{15} X \]
Thus,
\[ B_{NS} = \frac{9}{15} X - \frac{2}{15} X = \frac{7}{15} X \]

### Step 6: Calculate the Total Number of Boys
Now we can find the total number of boys:
\[ \text{Total boys} = B_S + B_{NS} = \frac{1}{5} X + \frac{7}{15} X \]
Finding a common denominator (15):
\[ \frac{1}{5} X = \frac{3}{15} X \]
Thus,
\[ \text{Total boys} = \frac{3}{15} X + \frac{7}{15} X = \frac{10}{15} X = \frac{2}{3} X \]

### Step 7: Find the Fraction of Boys Among First-Place Finishers
To find the fraction of the total number of first-place finishers who were boys:
\[ \text{Fraction of boys} = \frac{\text{Total boys}}{X} = \frac{\frac{2}{3} X}{X} = \frac{2}{3} \]

### Final Answer
The fraction of the total number of first-place finishers who were boys is:
\[ \frac{2}{3} \]