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During a rain shower, Jeanette collects $42\frac{1}{2}$ gallons of water in a rain barrel outside her home. She uses $1\frac{1}{4}$ gallons to water plants inside her house. Then, she uses $\frac{1}{2}$ of the remaining water to wash a load of dirty laundry. How many gallons of water does Jeanette use to wash her laundry?$\newline$Write your answer as a whole number, fraction, or mixed number. Simplify any fractions.$\newline$$\_\_\_$ gallons$\newline$