Question

Melinda can paint
1
4
of a wall in the same time that Desiree can paint
2
5
of a wall.

How much of a wall will Melinda have painted when Desiree has finished painting
1
wall?

Answer 1

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.

1. **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:
\[
\text{Rate of Desiree} = \frac{\frac{2}{5} \text{ wall}}{t \text{ hours}} = \frac{2}{5t} \text{ walls/hour}
\]

2. **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:
\[
\text{Time} = \frac{1 \text{ wall}}{\frac{2}{5t} \text{ walls/hour}} = \frac{5t}{2}
\]

3. **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.