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Let's solve the problem step by step.

The problem states that the final score in a gymnastics competition is calculated by taking 75% of the average technical score and adding 25% of the artistic score. 

We are given:
- The average technical score (\( T \)) = 7.6
- The required final score (\( F \)) = 8.0

We need to find the artistic score (\( A \)) that will give a final score of at least 8.0.

The formula for the final score (\( F \)) is:
\[
F = 0.75T + 0.25A
\]

Substituting the values we know:
\[
8.0 = 0.75(7.6) + 0.25A
\]

Let's calculate \( 0.75 \times 7.6 \) first:
\[
0.75 \times 7.6 = 5.7
\]

So the equation becomes:
\[
8.0 = 5.7 + 0.25A
\]

Now, isolate \( A \) by subtracting 5.7 from both sides:
\[
8.0 - 5.7 = 0.25A
\]
\[
2.3 = 0.25A
\]

Finally, solve for \( A \) by dividing both sides by 0.25:
\[
A = \frac{2.3}{0.25} = 9.2
\]

So, the gymnast needs an artistic score of **9.2** to achieve a final score of at least 8.0.

Would you like more details or have any questions?

Here are five related questions you could explore:
1. What happens if the technical score decreases? How does that affect the required artistic score?
2. How would the required artistic score change if the gymnast wants a final score of 8.5 instead?
3. What if the weights of technical and artistic scores were reversed? How would that change the calculations?
4. Can the gymnast still achieve an 8.0 if the technical score is lower than 7.6? What would the artistic score need to be?
5. How would rounding the artistic score to the nearest tenth affect the final score?

**Tip:** When dealing with weighted averages, always pay close attention to the percentages assigned to each component to ensure accurate calculations.

In a gymnastics competition, an athlete's final score is calculated by taking 75% of the average technical score and adding 25% of the artistic score. All scores are out of 10, and one gymnast has a 7.6 average technical score. What artistic score does the gymnast need to have a final score of at least 8.0?

Learn how to calculate the artistic score required in gymnastics to achieve a final score of at least 8.0, given a technical score of 7.6. This problem involves algebra and weighted averages, suitable for students in Grades 8-10.

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