To determine the relationship of the product \( \frac{2}{5} \times \frac{9}{7} \) to the two fractions, we can analyze the values:
- The value of \( \frac{2}{5} \) is 0.4.
- The value of \( \frac{9}{7} \) is approximately 1.2857.
When we multiply these two fractions: \[ \frac{2}{5} \times \frac{9}{7} = \frac{2 \times 9}{5 \times 7} = \frac{18}{35} \]
Now we can compare \( \frac{18}{35} \) to both \( \frac{2}{5} \) and \( \frac{9}{7} \).
-
To compare \( \frac{18}{35} \) and \( \frac{2}{5} \):
- Convert \( \frac{2}{5} \) to a fraction with a denominator of 35: \[ \frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \]
- Now \( \frac{18}{35} > \frac{14}{35} \), or \( \frac{18}{35} > \frac{2}{5} \).
-
To compare \( \frac{18}{35} \) and \( \frac{9}{7} \):
- Convert \( \frac{9}{7} \) to a fraction with a denominator of 35: \[ \frac{9}{7} = \frac{9 \times 5}{7 \times 5} = \frac{45}{35} \]
- Now \( \frac{18}{35} < \frac{45}{35} \), or \( \frac{18}{35} < \frac{9}{7} \).
Thus, the correct phrases to fill in the blanks are:
The product will be greater than \( \frac{2}{5} \) and less than \( \frac{9}{7} \).