Question

Jacob with his mower can mow and trim Mr. Clark's yard in 2 and a half hours, while Nicole with her mower can mow and trim the same yard in just 2 hours. If they work together on this yard and keep out of each other's way, in how many minutes can they complete the yard? Please show your work.

Answer 1

find the rate at which each worker works:

Jacob mows 1 yard in 2.5 hours, so he has a rate of 1/2.5 yards/hour
What is Nicole's rate?

To find the the time for both of them to do the job together, add their rates. \[\frac{1\text{ job}}{\text{combined rate}} = \text{ time for job}\]

Answer 2

Okay. So you're saying I should combine their rates of work; so Nicole can finish the yard in 2 hours. 2 plus 2/12 would be four and a half hours. If you divide that by 1, you get 0.22 or 22 minutes. Would this be correct?

Answer 3

*2 1/2

Answer 4

Let's do a sanity check on your answer. Jacob takes 2 1/2 hours to do the job. Nicole takes 2 hours to do the job. Together, they can do it in 22 minutes? Does that seem at all reasonable to you? If we assume that each one does about half of the work, that implies that they could do it themselves in less than an hour each!

Answer 5

Also, you do some imaginative math, but it isn't correct. 0.22 does not equal 22 minutes. \[0.22 \text{ hours} = 0.22 \cancel{\text{ hours}}*\frac{60 \text{ minutes}}{1\cancel{\text{ hour}}} = 13.2\text{ minutes}\]

Here's how you should have worked the problem:

Jacob's rate = \(\dfrac{1 \text{ yard}}{2\frac{1}{2}\text{ hours}} = \dfrac{2}{5}\dfrac{\text{ yard}}{\text{ hour}}\)
Nicole's rate = \(\dfrac{1 \text{ yard}}{2\text{ hours}} = \dfrac{1}{2}\dfrac{\text{ yard}}{\text{ hour}}\)

combined rate \(= (\dfrac{2}{5} + \dfrac{1}{2})\dfrac{\text{ yard}}{\text{ hour}} \)

Time to do the job together \( = \dfrac{1 \cancel{\text{ yard}}}{ (\dfrac{2}{5} + \dfrac{1}{2})\dfrac{\cancel{\text{ yard}}}{\text{ hour}} } = \text{__________ hour} \)

Hint: the answer is a bit more than an hour. This makes sense, because each of them does at most 1/2 of the yard in 1 hour, so after 1 hour, 1/2+2/5 = 9/10 of the yard will be done.

You can (and should!) check your answer by multiplying the total time by each worker's rate and adding the resulting quantities. If it doesn't add up to 1 job, you've made a mistake somewhere.