In an institute `60% ` of the students are boys and the rest are girls. Further `15% ` of the boys and `7.5% ` of the girls are getting a fee waiver. If the number of those getting a fee waiver is 90, find the total number of students getting `50% ` concessions if it is given that `50%` of those not getting a fee waiver are eligible to get half fee concession?

Answer

To solve the problem step by step, we will follow the given information and calculations:

### Step 1: Determine the total number of students
Let the total number of students in the institute be \( x \).

### Step 2: Calculate the number of boys and girls
Since 60% of the students are boys, the number of boys is:
\[ \text{Number of boys} = 0.6x \]
The remaining 40% are girls, so the number of girls is:
\[ \text{Number of girls} = 0.4x \]

### Step 3: Calculate the number of boys and girls getting a fee waiver
15% of the boys are getting a fee waiver, so the number of boys getting a fee waiver is:
\[ \text{Boys getting fee waiver} = 0.15 \times 0.6x = 0.09x \]
7.5% of the girls are getting a fee waiver, so the number of girls getting a fee waiver is:
\[ \text{Girls getting fee waiver} = 0.075 \times 0.4x = 0.03x \]

### Step 4: Calculate the total number of students getting a fee waiver
The total number of students getting a fee waiver is:
\[ \text{Total fee waivers} = 0.09x + 0.03x = 0.12x \]

### Step 5: Set up the equation based on the given information
We know that the total number of students getting a fee waiver is 90:
\[ 0.12x = 90 \]

### Step 6: Solve for \( x \)
To find \( x \), divide both sides by 0.12:
\[ x = \frac{90}{0.12} = 750 \]

### Step 7: Calculate the number of students not getting a fee waiver
The number of students not getting a fee waiver is:
\[ \text{Students not getting fee waiver} = x - 90 = 750 - 90 = 660 \]

### Step 8: Calculate the number of students eligible for 50% concession
According to the problem, 50% of those not getting a fee waiver are eligible for a 50% concession:
\[ \text{Students getting 50% concession} = 0.5 \times 660 = 330 \]

### Final Answer
The total number of students getting a 50% concession is:
\[ \boxed{330} \]