The given equation is \( 9m = 10n \). To analyze this equation in terms of proportions, we can rewrite it in the form that relates \( m \) and \( n \).
From the equation \( 9m = 10n \), we can isolate \( \frac{m}{n} \):
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Divide both sides by \( n \) (assuming \( n \neq 0 \)): \[ \frac{9m}{n} = 10 \]
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Then divide both sides by 9: \[ \frac{m}{n} = \frac{10}{9} \]
This indicates that:
- \( \frac{m}{n} = \frac{10}{9} \)
Now, applying this further, we can also express it in the inversed proportion:
- Taking the reciprocal gives: \[ \frac{n}{m} = \frac{9}{10} \]
Now let's revisit the options provided in the question:
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\( \frac{9}{n} = \frac{10}{m} \)
- Not correct, because we just derived \( \frac{m}{n} = \frac{10}{9} \).
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\( \frac{9}{m} = \frac{10}{n} \)
- Not correct, follows the same reasoning.
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\( \frac{9}{10} = \frac{m}{n} \)
- Not correct, because \( \frac{m}{n} = \frac{10}{9} \).
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\( \frac{10}{9} = \frac{n}{m} \)
- Not correct, as it should be reversed (it should be \( \frac{n}{m} = \frac{9}{10} \)).
Hence, based on the analysis, the correct proportions derived from \( 9m = 10n \) are:
- \( \frac{m}{n} = \frac{10}{9} \)
- \( \frac{n}{m} = \frac{9}{10} \)
So out of the options presented, none seem to correctly represent the relationships as derived from the equation \( 9m = 10n \).