Question

Which fraction makes the statement%C2%A0TRUE? 210< %C2%A0 %C2%A0<711

Answer 1

To determine which fraction makes the statement \( 210 < \frac{x}{y} < 711 \) true, we can set two inequalities based on the endpoints given.

1. **First Inequality**:
\[
210 < \frac{x}{y}
\]
This can be rewritten as:
\[
210y < x
\]

2. **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.