Let's analyze each equation one by one to see which ones are true for the number 9.
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4 divide by ____ = 9/4 To find the missing number, we set up the equation: \[ \frac{4}{x} = \frac{9}{4} \] Cross-multiplying gives: \[ 4 \cdot 4 = 9 \cdot x \implies 16 = 9x \implies x = \frac{16}{9} \] This equation is true if \(x = \frac{16}{9}\).
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____ 1/9 = ____ divide by 9 Let's set it up. Let \(a\) be the first blank and \(b\) be the second blank: \[ \frac{a}{9} = \frac{b}{9} \] From this, we see that \(a = b\). However, we can't determine specific values for \(a\) and \(b\) just from the equation. But for \(a = b\), if we choose \(a = 1\) and \(b = 1\), the equation holds true. So, this equation can be true for certain values.
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____ 7 divide by _____ = 7/9 Let’s denote the blanks as \(c\) and \(d\): \[ \frac{c}{d} = \frac{7}{9} \] Rearranging gives: \[ 9c = 7d \implies c = \frac{7d}{9} \] This equation is true if \(c\) and \(d\) maintain the ratio of \(7:9\). For example, if \(d = 9\), then \(c = 7\).
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___ ___ divide by 1/9 = 81 Denote the two blanks as \(e\) and \(f\): \[ \frac{e}{1/9} = 81 \] This can be rewritten as: \[ e \cdot 9 = 81 \implies e = \frac{81}{9} = 9 \] So, if \(e = 9\), then the equation holds true.
From this analysis, the following equations are true for the number 9:
- 4 divide by ____ = 9/4 (True if the blank is 16/9)
- ____ 1/9 = ____ divide by 9 (True if both blanks are equal)
- ____ 7 divide by _____ = 7/9 (True if the blanks maintain the ratio 7:9)
- ___ ___ divide by 1/9 = 81 (True if one blank is 9)
All equations have valid interpretations that can include the number 9. Thus, all four statements are potentially true based on the specified constraints.