To apply the rule "multiply by the reciprocal" for the expression \( 5 \div \frac{2}{9} \), we first convert the division into multiplication by the reciprocal of \(\frac{2}{9}\).
The reciprocal of \(\frac{2}{9}\) is \(\frac{9}{2}\). Therefore,
\[ 5 \div \frac{2}{9} = 5 \times \frac{9}{2} \]
Now, let's write \( 5 \) as a fraction:
\[ \frac{5}{1} \times \frac{9}{2} \]
Now, multiplying these fractions:
\[ \frac{5 \times 9}{1 \times 2} = \frac{45}{2} \]
Among the options you provided:
- \( \frac{5}{1} \times \frac{9}{2} = \frac{45}{2} \) ✔️
- \( \frac{5}{1} \times \frac{2}{9} = \frac{10}{9} \) ❌ (this is incorrect as it is not using the reciprocal)
- \( \frac{1}{5} \times \frac{9}{2} = \frac{9}{10} \) ❌ (this does not represent the original problem)
- \( \frac{1}{5} \times \frac{2}{9} = \frac{2}{45} \) ❌ (this also does not represent the original problem)
Therefore, the correct way to use the rule "multiply by the reciprocal" for the expression is:
\( \frac{5}{1} \times \frac{9}{2} = \frac{45}{2} \)