To determine which fraction makes the statement \( 210 < \frac{x}{y} < 711 \) true, we can set two inequalities based on the endpoints given.
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First Inequality: \[ 210 < \frac{x}{y} \] This can be rewritten as: \[ 210y < x \]
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Second Inequality: \[ \frac{x}{y} < 711 \] This can be rewritten as: \[ x < 711y \]
Combining these inequalities, we have: \[ 210y < x < 711y \]
Now we can choose any values for \(y\) (with \(y > 0\)) and find an appropriate value of \(x\) that satisfies both inequalities.
Example: Let’s take \(y = 1\):
- For \(y = 1\):
- From the first inequality: \(210 < x\)
- From the second inequality: \(x < 711\)
So, for \(y = 1\): \[ 210 < x < 711 \]
You can choose any integer or fraction for \(x\) that falls between 210 and 711, such as:
- \(x = 300\), giving the fraction \(\frac{300}{1} = 300\)
- \(x = 500\), giving the fraction \(\frac{500}{1} = 500\)
- Or, with \(x = 400\) and \(y = 2\), \(x = 400\) would give \(\frac{400}{2} = 200\) which doesn't satisfy the range.
Thus, any fraction of the form \(\frac{x}{y}\) where \(210y < x < 711y\) is valid. You can choose \(y\) as long as \(y > 0\) and find suitable \(x\).
In summary, appropriate fractions such as \(\frac{300}{1}\), \(\frac{500}{1}\), or \(\frac{711}{1}\) can be chosen to make the statement true.