To find out how many days the bananas will last, we need to divide the total amount of bananas by the amount the elephants eat each day.
- Total buckets of bananas: \( 4 \frac{1}{2} = \frac{9}{2} \) buckets
- Daily consumption: \( \frac{3}{4} \) of a bucket per day
Now, we will perform the division:
\[ \text{Number of days} = \frac{\text{Total buckets}}{\text{Daily consumption}} = \frac{\frac{9}{2}}{\frac{3}{4}} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{9}{2} \div \frac{3}{4} = \frac{9}{2} \times \frac{4}{3} \]
Now, we can simplify:
\[ = \frac{9 \times 4}{2 \times 3} = \frac{36}{6} = 6 \]
Thus, the bananas will last for 6 days.