Question

Hayden is a manager at a landscaping company. He has two workers to landscape an entire park, Cody and Kaitlyn. Cody can complete the project in 8 hours. Kaitlyn can complete the project in 6 hours. Hayden wants to know how long it will take them to complete the project together.
Write an equation and solve for the time it takes Cody and Kaitlyn to complete the project together. Explain each step.

Answer 1

Answer: 3.4 h

Explanation:

1) The basis to solve this kind of problems is that the speed of working together is equal to the sum of the individual speeds.

This is: speed of doing the project together = speed of Cody working alone + speed of Kaitlyin working alone.

2) Speed of Cody

Cody can complete the project in 8 hours => 1 project / 8 h

3) Speed of Kaitlyn

Kaitlyn can complete the project in 6 houres => 1 project / 6 h

4) Speed working together:

1 / 8 + 1 / 6  = [6 + 8] / (6*8  = 14 / 48 = 7 / 24

7/24 is the velocity or working together meaning that they can complete 7 projects in 24 hours.

Then, the time to complete the entire project is the inverse: 24 hours / 7 projects ≈ 3.4 hours / project.Meaning 3.4 hours to complete the project.

Answer 2

Denote all work that should be done as 1. Then if Cody can complete the project in 8 hours, he can do $\frac{1}{8}$ per hour and if Kaitlyn can complete the project in 6 hours, then she can do $\frac{1}{6}$ per hour. Together they complete $\frac{1}{8}+\frac{1}{6}$ per hour.

Simplify this expression:

$\dfrac{1}{8} +\dfrac{1}{6} =\dfrac{3}{24} +\dfrac{4}{24}=\dfrac{7}{24}$.

Then it will take them to complete the project together $\dfrac{1}{\frac{7}{24}} =\dfrac{24}{7} =3\dfrac{3}{7}$ hours

Answer: It will take them to complete the project together $3\dfrac{3}{7}$ hours.