To understand why the solutions to the three proportions \( \frac{36}{4} = \frac{x}{7} \), \( \frac{4}{36} = \frac{7}{x} \), and \( \frac{36}{x} = \frac{4}{7} \) are the same, we will analyze each proportion step by step.
Proportion 1: \( \frac{36}{4} = \frac{x}{7} \)
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Cross-multiply: \[ 36 \cdot 7 = 4 \cdot x \] This simplifies to: \[ 252 = 4x \]
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Solve for \( x \): \[ x = \frac{252}{4} = 63 \]
Proportion 2: \( \frac{4}{36} = \frac{7}{x} \)
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Cross-multiply: \[ 4x = 36 \cdot 7 \] This simplifies to: \[ 4x = 252 \]
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Solve for \( x \): \[ x = \frac{252}{4} = 63 \]
Proportion 3: \( \frac{36}{x} = \frac{4}{7} \)
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Cross-multiply: \[ 36 \cdot 7 = 4 \cdot x \] This simplifies to: \[ 252 = 4x \]
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Solve for \( x \): \[ x = \frac{252}{4} = 63 \]
Conclusion
In each case, we end up with the same equation, \( 4x = 252 \), irrespective of how we set up the proportion. Each proportion represents the same relationship between the values involved. Therefore, we find that the solutions for \( x \) from all three proportions are equal, yielding \( x = 63 \) consistently across each proportion.
This demonstrates that the ratios are equivalent; hence, the value of \( x \) remains unchanged across different proportional setups.