To find the total cost of the potting soil that Arun needs to buy, we first need to calculate the total weight of the potting soil in pounds, which is given as \(8 \frac{3}{4}\) pounds.
First, we convert \(8 \frac{3}{4}\) to an improper fraction:
\[ 8 \frac{3}{4} = 8 + \frac{3}{4} = \frac{8 \cdot 4}{4} + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \text{ pounds} \]
Next, we calculate the total cost. The cost per pound is $0.60, so we multiply the total weight in pounds by the cost per pound:
\[ \text{Total Cost} = \text{Weight in pounds} \times \text{Cost per pound} = \frac{35}{4} \times 0.60 \]
Now, calculating this step-by-step:
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First, convert 0.60 to a fraction: \[ 0.60 = \frac{60}{100} = \frac{3}{5} \]
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Now perform the multiplication: \[ \text{Total Cost} = \frac{35}{4} \times \frac{3}{5} = \frac{35 \times 3}{4 \times 5} = \frac{105}{20} \]
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Simplify the fraction: \[ \frac{105}{20} = \frac{21}{4} \text{ dollars} \]
Finally, converting \(\frac{21}{4}\) into a decimal or mixed number:
\[ \frac{21}{4} = 5.25 \text{ dollars} \]
Thus, the total cost of the potting soil that Arun should buy to fill the flowerpot is:
\[ \boxed{5.25} \] dollars.