Let's denote Dirk's cleaning rate as [tex]\(D\)[/tex] and Geralyn's cleaning rate as [tex]\(G\)[/tex]. Here are their rates:
- Dirk can clean the playground in 5 hours, which means his rate is [tex]\(\frac{1 \text{ playground}}{5 \text{ hours}} = \frac{1}{5}\)[/tex] of a playground per hour.
- Geralyn can clean the playground in 4 hours, which means her rate is [tex]\(\frac{1 \text{ playground}}{4 \text{ hours}} = \frac{1}{4}\)[/tex] of a playground per hour.
When they work together, their combined rate is simply the sum of their individual rates. Let's denote the total time it takes for them to clean the playground together as [tex]\(x\)[/tex] hours. This combined rate can be represented as [tex]\(\frac{1}{x}\)[/tex] of a playground per hour.
Our goal is to find an equation that represents this situation. So, we sum their rates:
[tex]\[
\frac{1}{5} + \frac{1}{4} = \frac{1}{x}
\][/tex]
Now, let's review the options provided:
1. [tex]\(\frac{1}{5} + \frac{1}{x} = \frac{1}{4}\)[/tex]
2. [tex]\(\frac{1}{x} + \frac{1}{4} = \frac{1}{5}\)[/tex]
3. [tex]\(\frac{1}{5} + \frac{1}{4} = \frac{1}{x}\)[/tex]
4. [tex]\(\frac{1}{4} + \frac{1}{5} = \frac{x}{9}\)[/tex]
The third equation, [tex]\(\frac{1}{5} + \frac{1}{4} = \frac{1}{x}\)[/tex], correctly represents the scenario where Dirk and Geralyn are working together to clean the playground. Therefore, the correct answer is:
[tex]\[
\frac{1}{5} + \frac{1}{4} = \frac{1}{x}
\][/tex]
Finally, solving this equation for [tex]\(x\)[/tex], we find that the total time it would take for Dirk and Geralyn to clean the playground together is roughly 2.22 hours.
