To determine which ratio results in a valid proportion when set equal to \( \frac{7}{3} \), we need to check if the cross products of the fractions are equal. A proportion states that two ratios are equal, so if we set \( \frac{7}{3} = \frac{a}{b} \), then we must have:
\[ 7b = 3a \]
Now, let's evaluate each option:
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Option 1: \( \frac{9}{49} \)
\[ 7 \cdot 49 \quad \text{and} \quad 3 \cdot 9 \] \[ 343 \quad \text{and} \quad 27 \] \( 343 \neq 27 \) (not proportional)
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Option 2: \( \frac{18}{42} \)
\[ 7 \cdot 42 \quad \text{and} \quad 3 \cdot 18 \] \[ 294 \quad \text{and} \quad 54 \] \( 294 \neq 54 \) (not proportional)
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Option 3: \( \frac{42}{18} \)
\[ 7 \cdot 18 \quad \text{and} \quad 3 \cdot 42 \] \[ 126 \quad \text{and} \quad 126 \] \( 126 = 126 \) (proportional)
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Option 4: \( \frac{49}{9} \)
\[ 7 \cdot 9 \quad \text{and} \quad 3 \cdot 49 \] \[ 63 \quad \text{and} \quad 147 \] \( 63 \neq 147 \) (not proportional)
Thus, the ratio that results in a valid proportion with \( \frac{7}{3} \) is:
\[ \frac{42}{18} \]