To determine how many days 7 and 1/2 bags of dog food will last when using 5/6 of a bag per day, we can set up the following equation:
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Convert 7 and 1/2 bags to an improper fraction: \[ 7 \frac{1}{2} = \frac{15}{2} \]
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Let \( x \) be the number of days the dog food will last. The amount of dog food consumed in \( x \) days is given by: \[ \text{Dog food consumed} = \frac{5}{6} \times x \]
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Set up the equation to find \( x \): \[ \frac{5}{6} x = \frac{15}{2} \]
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To solve for \( x \), multiply both sides of the equation by the reciprocal of \(\frac{5}{6}\): \[ x = \frac{15}{2} \times \frac{6}{5} \]
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Perform the multiplication: \[ x = \frac{15 \times 6}{2 \times 5} = \frac{90}{10} = 9 \]
Thus, 7 and 1/2 bags of dog food will last for \(\boxed{9}\) days.