Suppose that a lawn can be raked by one gardener in 5 hours and by a second gardener in 4 hours. How long it will take the two gardeners to rake the lawn working ogether? Norking togethor

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem: Calculating Combined Work Time**

*Scenario*

Suppose that a lawn can be raked by one gardener in 5 hours and by a second gardener in 4 hours. The question is: How long will it take for the two gardeners to rake the lawn working together?

*Solution Approach*

When two or more people are working together to complete a task, we can add their rates of work to find the total time required to complete the task.

1. **Gardener 1's Rate:**  
   One garden in 5 hours → Work rate = \( \frac{1}{5} \) of the garden per hour.

2. **Gardener 2's Rate:**  
   One garden in 4 hours → Work rate = \( \frac{1}{4} \) of the garden per hour.

3. **Combined Rate:**  
   Combined work rate = \( \frac{1}{5} + \frac{1}{4} \)

   To calculate, find a common denominator (20):  
   \[
   \frac{1}{5} = \frac{4}{20} \\
   \frac{1}{4} = \frac{5}{20} \\
   \]

   Add the rates:  
   \[
   \frac{4}{20} + \frac{5}{20} = \frac{9}{20}
   \]

4. **Total Time:**  
   To find the time, take the reciprocal of the combined rate:  
   \[
   \text{Time} = \frac{20}{9} \approx 2.22
   \]

*Solution*

Working together, the two gardeners can rake the garden in about 2.22 hours. (Round to two decimal places as needed.)

*Answer Box:* Enter your answer in the box.
Transcribed Image Text:**Problem: Calculating Combined Work Time** *Scenario* Suppose that a lawn can be raked by one gardener in 5 hours and by a second gardener in 4 hours. The question is: How long will it take for the two gardeners to rake the lawn working together? *Solution Approach* When two or more people are working together to complete a task, we can add their rates of work to find the total time required to complete the task. 1. **Gardener 1's Rate:** One garden in 5 hours → Work rate = \( \frac{1}{5} \) of the garden per hour. 2. **Gardener 2's Rate:** One garden in 4 hours → Work rate = \( \frac{1}{4} \) of the garden per hour. 3. **Combined Rate:** Combined work rate = \( \frac{1}{5} + \frac{1}{4} \) To calculate, find a common denominator (20): \[ \frac{1}{5} = \frac{4}{20} \\ \frac{1}{4} = \frac{5}{20} \\ \] Add the rates: \[ \frac{4}{20} + \frac{5}{20} = \frac{9}{20} \] 4. **Total Time:** To find the time, take the reciprocal of the combined rate: \[ \text{Time} = \frac{20}{9} \approx 2.22 \] *Solution* Working together, the two gardeners can rake the garden in about 2.22 hours. (Round to two decimal places as needed.) *Answer Box:* Enter your answer in the box.
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