Questions: If 85% of the girls in a class wanted to have a prom and 71% of the boys wanted a prom, is it possible that only 58% of the students in the class wanted a prom? Explain your answer. Choose the correct answer below. A. No, it is not possible. Since more than 58% of boys and more than 58% of girls wanted a prom, more than 58% of all students must have wanted to have a prom. B. Yes, it is possible. If the number of boys in the class was equal to the number of girls, then it would be possible that only 58% of the students wanted to have a prom. C. Yes, it is possible. If there were enough more boys than girls in the class, then it would be possible that only 58% of the students wanted to have a prom. D. No, it is not possible. Since a lesser percentage of boys than girls wanted to have a prom, less than 58% of all students must have wanted to have a prom. E. Yes, it is possible. If there were enough more girls than boys in the class, then it would be possible that only 58% of the students wanted to have a prom.

If 85% of the girls in a class wanted to have a prom and 71% of the boys wanted a prom, is it possible that only 58% of the students in the class wanted a prom? Explain your answer. Choose the correct answer below. A. No, it is not possible. Since more than 58% of boys and more than 58% of girls wanted a prom, more than 58% of all students must have wanted to have a prom. B. Yes, it is possible. If the number of boys in the class was equal to the number of girls, then it would be possible that only 58% of the students wanted to have a prom. C. Yes, it is possible. If there were enough more boys than girls in the class, then it would be possible that only 58% of the students wanted to have a prom. D. No, it is not possible. Since a lesser percentage of boys than girls wanted to have a prom, less than 58% of all students must have wanted to have a prom. E. Yes, it is possible. If there were enough more girls than boys in the class, then it would be possible that only 58% of the students wanted to have a prom.
Transcript text: If $85 \%$ of the girls in a class wanted to have a prom and $71 \%$ of the boys wanted a prom, is it possible that only $58 \%$ of the students in the class wanted a prom? Explain your answer. Choose the correct answer below. A. No, it is not possible. Since more than $58 \%$ of boys and more than $58 \%$ of girls wanted a prom, more than $58 \%$ of all students must have wanted to have a prom. B. Yes, it is possible. If the number of boys in the class was equal to the number of girls, then it would be possible that only $58 \%$ of the students wanted to have a prom. C. Yes, it is possible. If there were enough more boys than girls in the class, then it would be possible that only $58 \%$ of the students wanted to have a prom. D. No, it is not possible. Since a lesser percentage of boys than girls wanted to have a prom, less than $58 \%$ of all students must have wanted to have a prom. E. Yes, it is possible. If there were enough more girls than boys in the class, then it would be possible that only $58 \%$ of the students wanted to have a prom
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Solution

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Solution Steps

To determine if it is possible that only 58% of the students in the class wanted a prom, given that 85% of the girls and 71% of the boys wanted a prom, we need to consider the weighted average of the percentages based on the number of boys and girls in the class. We can set up an equation to check if there exists a ratio of boys to girls that satisfies the given condition.

Step 1: Set Up the Equation

We need to determine if it is possible for 58% of the students to want a prom given that 85% of the girls and 71% of the boys want a prom. We set up the equation for the weighted average of the percentages:

\[ \frac{0.85x + 0.71y}{x + y} = 0.58 \]

where \( x \) is the number of girls and \( y \) is the number of boys.

Step 2: Solve for the Ratio of Boys to Girls

We solve the equation for \( y \) in terms of \( x \):

\[ 0.85x + 0.71y = 0.58(x + y) \]

Simplifying, we get:

\[ 0.85x + 0.71y = 0.58x + 0.58y \]

\[ 0.85x - 0.58x = 0.58y - 0.71y \]

\[ 0.27x = -0.13y \]

\[ y = -\frac{0.27}{0.13}x \]

\[ y = -2.0769x \]

Step 3: Interpret the Solution

The solution \( y = -2.0769x \) indicates that the number of boys \( y \) would need to be negative for any positive number of girls \( x \). This is not possible in a real-world scenario because the number of students cannot be negative.

Final Answer

Since the number of boys cannot be negative, it is not possible for only 58% of the students to want a prom given the percentages of boys and girls who want a prom.

The answer is:

\[ \boxed{\text{A}} \]

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