An opinion poll survey was conducted in which 120 sports fans and 75 non-sports fans participated. If the sample size was increased by 65 non-sports fans, how many sports fans should be added so that `(3)/(5)` of those polled are sports fans?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the initial conditions
We start with:
- Number of sports fans = 120
- Number of non-sports fans = 75

### Step 2: Calculate the new number of non-sports fans
If we increase the sample size by 65 non-sports fans, the new number of non-sports fans will be:
\[ \text{New non-sports fans} = 75 + 65 = 140 \]

### Step 3: Set up the equation for the total participants
Let \( x \) be the number of sports fans we need to add. The new number of sports fans will then be:
\[ \text{New sports fans} = 120 + x \]
The total number of participants will be:
\[ \text{Total participants} = \text{New sports fans} + \text{New non-sports fans} = (120 + x) + 140 = 260 + x \]

### Step 4: Set up the ratio equation
We want \( \frac{3}{5} \) of the total participants to be sports fans. Therefore, we can set up the equation:
\[ \frac{120 + x}{260 + x} = \frac{3}{5} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(120 + x) = 3(260 + x) \]

### Step 6: Expand both sides
Expanding both sides results in:
\[ 600 + 5x = 780 + 3x \]

### Step 7: Rearrange the equation
Now, rearranging the equation to isolate \( x \):
\[ 5x - 3x = 780 - 600 \]
This simplifies to:
\[ 2x = 180 \]

### Step 8: Solve for \( x \)
Dividing both sides by 2 gives:
\[ x = \frac{180}{2} = 90 \]

### Conclusion
Thus, the number of sports fans that should be added is:
\[ \boxed{90} \]

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