To determine what "All-Around" scores the gymnast needs in her next competition to achieve an average score of at least 36 across all competitions, we first need to calculate the total score required for her to reach this average.
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Calculate the current total score:
The gymnast's current scores are: 37.7, 35.7, 38.5, and 37.\[ \text{Current Total Score} = 37.7 + 35.7 + 38.5 + 37 = 149.9 \]
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Calculate the number of competitions so far:
She has competed in 4 competitions. -
Determine the total number of competitions after the next one:
After the next competition, she will have participated in 5 competitions. -
Calculate the total score needed to have an average of 36 after 5 competitions:
To find the total score required for an average of 36, we multiply the desired average by the total number of competitions:\[ \text{Total Score Needed} = 36 \times 5 = 180 \]
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Calculate the required score in the next competition:
Let \( x \) be the score she needs in the next competition. Therefore, we set up the following equation:\[ \text{Current Total Score} + x = \text{Total Score Needed} \]
Substituting in the values:
\[ 149.9 + x = 180 \]
Now, solve for \( x \):
\[ x = 180 - 149.9 = 30.1 \]
Therefore, the gymnast needs to score at least 30.1 in her next competition to achieve an average of 36 or higher and be eligible to move up to the next level. Since scores in gymnastics are often rounded, she would need to score at least 31 to ensure she meets the requirement.