To find out how much of a wall Melinda will have painted when Desiree has finished painting 1 wall, we can first determine their painting rates.
Melinda can paint \(\frac{1}{4}\) of a wall in the same time Desiree paints \(\frac{2}{5}\) of a wall.
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Find the rate of Desiree: Let's denote the time it takes for Desiree to paint \(\frac{2}{5}\) of a wall as \(t\). In that time, she can complete: \[ \text{Rate of Desiree} = \frac{\frac{2}{5} \text{ wall}}{t \text{ hours}} = \frac{2}{5t} \text{ walls/hour} \]
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Calculate the time taken by Desiree to paint 1 wall: To find out how long it takes her to paint 1 wall: \[ 1 \text{ wall} = \text{Rate of Desiree} \times \text{time} \] Rearranging for time when painting 1 wall: \[ \text{Time} = \frac{1 \text{ wall}}{\frac{2}{5t} \text{ walls/hour}} = \frac{5t}{2} \]
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Find how much Melinda can paint in that same time: In the time \(\frac{5t}{2}\), we can calculate how much Melinda can paint. We know Melinda's rate per hour is derived from her painting \(\frac{1}{4}\) of a wall in the time \(t\) (which is the same time Desiree painted \(\frac{2}{5}\)): \[ \text{Rate of Melinda} = \frac{\frac{1}{4} \text{ wall}}{t} = \frac{1}{4t} \text{ walls/hour} \] Now, we can find out how much Melinda can paint in \(\frac{5t}{2}\): \[ \text{Melinda's painted wall} = \text{Rate of Melinda} \times \text{Time} = \left(\frac{1}{4t}\right) \cdot \left(\frac{5t}{2}\right) \] \[ = \frac{5t}{8t} = \frac{5}{8} \text{ walls} \]
So, when Desiree has finished painting 1 wall, Melinda will have painted \(\frac{5}{8}\) of a wall.