Question

question 3 of 5
select the correct answer.
the ratio of adults to students on a class field trip is 2 to 5. if there are 12 more students than adults, which proportion and solution represent this situation?
\\(\frac{2}{5} = \frac{x + 12}{x}; x = 20\\)
\\(\frac{2}{5} = \frac{x + 12}{x}; x = 4\\)
\\(\frac{2}{5} = \frac{x}{x + 12}; x = 8\\)
\\(\frac{2}{5} = \frac{x}{x + 12}; x = 4\\)

Explanation:

Step1: Define variables

Let \( x \) be the number of adults. Then the number of students is \( x + 12 \) (since there are 12 more students than adults).

Step2: Set up the proportion

The ratio of adults to students is \( 2:5 \), so \( \frac{\text{Adults}}{\text{Students}}=\frac{2}{5} \). Substituting the variables, we get \( \frac{2}{5}=\frac{x}{x + 12} \).

Step3: Solve the proportion

Cross - multiply: \( 2(x + 12)=5x \)
Expand: \( 2x+24 = 5x \)
Subtract \( 2x \) from both sides: \( 24=5x - 2x \)
Simplify: \( 3x = 24 \)
Divide both sides by 3: \( x=\frac{24}{3}=8 \)? Wait, no, wait. Wait, let's check the options again. Wait, maybe I mixed up adults and students. Wait, the ratio of adults to students is 2 to 5, so adults : students = 2:5. If adults are \( x \), students are \( x + 12 \), so \( \frac{2}{5}=\frac{x}{x + 12} \). But let's check the options. Wait, the third option is \( \frac{2}{5}=\frac{x}{x + 12};x = 8 \)? Wait, no, let's solve \( \frac{2}{5}=\frac{x}{x + 12} \)
Cross - multiply: \( 2(x + 12)=5x\)
\( 2x+24 = 5x\)
\( 24 = 3x\)
\( x = 8 \). Wait, but let's check the first option: \( \frac{2}{5}=\frac{x + 12}{x} \). Let's solve that. Cross - multiply: \( 2x=5(x + 12) \)
\( 2x=5x + 60 \)
\( - 3x=60 \)
\( x=-20 \), which is not possible. Wait, I think I made a mistake in variable definition. Let's re - define: Let \( x \) be the number of adults, then students are \( x + 12 \). The ratio of adults to students is 2:5, so \( \frac{\text{Adults}}{\text{Students}}=\frac{2}{5}\), so \( \frac{x}{x + 12}=\frac{2}{5} \). But the third option is \( \frac{2}{5}=\frac{x}{x + 12};x = 8 \)? Wait, no, let's plug \( x = 8 \) into \( \frac{x}{x + 12} \), we get \( \frac{8}{20}=\frac{2}{5} \), which is correct. Wait, but let's check the first option: \( \frac{2}{5}=\frac{x + 12}{x} \). If \( x = 8 \), \( \frac{8 + 12}{8}=\frac{20}{8}=\frac{5}{2}
eq\frac{2}{5} \). Wait, maybe I had the ratio reversed. The ratio of adults to students is 2 to 5, so adults : students = 2:5, so students : adults=5:2. Wait, the problem says "the ratio of adults to students on a class field trip is 2 to 5". So adults (A) : students (S)=2:5, so \( \frac{A}{S}=\frac{2}{5} \). If there are 12 more students than adults, \( S=A + 12 \). Let \( A=x \), then \( S=x + 12 \), so \( \frac{x}{x + 12}=\frac{2}{5} \). Solving: \( 5x=2(x + 12)\), \( 5x=2x + 24 \), \( 3x=24 \), \( x = 8 \). So the proportion is \( \frac{2}{5}=\frac{x}{x + 12} \) and \( x = 8 \), which is the third option.

Answer:

\(\frac{2}{5}=\frac{x}{x + 12};x = 8\) (the third option)