If the fractions below are equivalent, what are the possible values for \( x \) and \( y \) ? \( \begin{array}{l}\frac{16}{36}, \frac{w}{y} \\ x=4, y=6 \\ x=12, y=32 \\ x=20, y=42 \\ x=24, y=54\end{array} \)

To determine the possible values for \( x \) and \( y \) given that the fractions are equivalent, we can start by simplifying the fraction \( \frac{16}{36} \). Dividing both the numerator and the denominator by their greatest common divisor, which is 4, we get: \[ \frac{16 \div 4}{36 \div 4} = \frac{4}{9} \] Now we are looking for values of \( y \) such that \( \frac{w}{y} = \frac{4}{9} \) holds true, where \( w \) represents \( x \) from the provided options. Let's analyze the options: 1. If \( x = 4 \): \( w = 4 \), we must check if \( \frac{4}{y} = \frac{4}{9} \). This gives \( y = 9 \), which does not match \( y = 6 \). 2. If \( x = 12 \): \( w = 12 \) implies \( \frac{12}{y} = \frac{4}{9} \). Cross-multiplying gives \( 12 \times 9 = 4y \) or \( 108 = 4y \), leading to \( y = 27 \), which does not match \( y = 32 \). 3. If \( x = 20 \): \( w = 20 \), we check \( \frac{20}{y} = \frac{4}{9} \). Cross-multiplying, \( 20 \times 9 = 4y \), gives \( 180 = 4y \), thus \( y = 45 \), not matching \( y = 42 \). 4. If \( x = 24 \): \( w = 24 \), we find \( \frac{24}{y} = \frac{4}{9} \). Cross-multiplication yields \( 24 \times 9 = 4y \), so \( 216 = 4y \), leading to \( y = 54 \), which aligns with \( y = 54 \). Thus, the only combination where \( w = x \) and \( y \) are such that the fractions are equivalent is \( x = 24 \) and \( y = 54 \).