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To solve this problem, let's follow these steps:

### Given:
- The opponent's score: **40.8 points**
- Degree of difficulty for both you and your opponent: **2.4**
- Two of your scores from the judges: **4 ½** (or 4.5) and **6 ½** (or 6.5)
- You need to find the score from the third judge that would give you more total points than your opponent.

### Step 1: Calculate the opponent's raw score
The total score is calculated by summing the judges' scores and multiplying by the degree of difficulty. So for the opponent:
\[
\text{Opponent's total score} = \text{sum of opponent's scores} \times \text{degree of difficulty}
\]
Let’s denote the sum of the opponent's scores as \( S_{\text{opponent}} \). Then:
\[
S_{\text{opponent}} \times 2.4 = 40.8
\]
Solve for \( S_{\text{opponent}} \):
\[
S_{\text{opponent}} = \frac{40.8}{2.4} = 17
\]

So, the sum of the opponent’s scores is **17**.

### Step 2: Calculate the current sum of your scores
You have two scores from the judges: **4.5** and **6.5**. The sum of these is:
\[
S_{\text{your current scores}} = 4.5 + 6.5 = 11
\]

### Step 3: Determine the necessary score from Judge 3
Let \( x \) be the score from Judge 3. Then the sum of your scores will be \( 11 + x \). To win, the total score from your judges must be more than 17. So:
\[
11 + x > 17
\]
Solve for \( x \):
\[
x > 17 - 11
\]
\[
x > 6
\]

### Step 4: Conclusion
The score from Judge 3 must be **greater than 6** for you to get more points than your opponent.

Would you like any further clarification or a more detailed explanation of any part? Here are five related questions you could explore:

1. How would the required score change if the degree of difficulty were higher?
2. What if one judge's score were unusually low? How would that affect the total score?
3. How would you adjust the calculation if there were only two judges?
4. Can a higher degree of difficulty compensate for a low score from one judge?
5. How would the math change if the scores had decimals, like 4.75 or 6.25?

**Tip:** Always make sure to account for all known variables before solving for unknowns!

In a diving competition, scores range from 0 to 10. When there are three judges, the number of points awarded is calculated by multiplying the sum of the scores by the degree of difficulty. Your opponent earns 40.8 points on a dive. You perform a dive with a degree of difficulty of 2.4. Two of your scores are shown: 4.5 and 6.5. What score from the third judge will give you more points than your opponent?

This math problem asks how to determine the score needed from a third judge in a diving competition to earn more points than an opponent. It involves calculating scores from judges and using algebraic inequalities to find the solution. Suitable for students in Grades 7-10.

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