Question

anna asked her trainee, kevin, to help call some of her clients to sell a newly-issued municipal bond. if she had to call all of the clients herself, it would take anna six hours to complete the job. if kevin alone were to call the clients, it would take him nine hours.

working together, how long will it take anna and kevin to call all of the clients?

a.) 4.5 hours
b.) 2.8 hours
c.) 3.0 hours
d.) 3.6 hours

Explanation:

Identify individual work rates

Let the total job of calling all clients be represented as \(1\) whole job.

• Anna's rate of work is \(\frac{1}{6}\) of the job per hour, since she takes \(6\) hours to complete it alone.
• Kevin's rate of work is \(\frac{1}{9}\) of the job per hour, since he takes \(9\) hours to complete it alone.

Determine the combined work rate

When working together, their individual rates are added to find their combined rate per hour:

$$ \text{Combined Rate} = \frac{1}{6} + \frac{1}{9} $$

To add these fractions, find a common denominator, which is \(18\):

$$ \frac{1}{6} = \frac{3}{18} $$

$$ \frac{1}{9} = \frac{2}{18} $$

$$ \text{Combined Rate} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18} \text{ of the job per hour} $$

Calculate the total time required

Let \(t\) be the time in hours it takes for them to complete the job together. The relationship between rate, time, and work is:

$$ \text{Rate} \times \text{Time} = \text{Work} $$

$$ \frac{5}{18} \cdot t = 1 $$

Solve for \(t\):

$$ t = \frac{18}{5} = 3.6 \text{ hours} $$

Answer:

• a.) 4.5 hours
• b.) 2.8 hours
• c.) 3.0 hours
• d.) 3.6 hours (Correct answer)