Classic combined rates of work word Problem

[LearninDaMath]

This is not a homework question. It is a question my sister asked me, and although I could solve it, I could not explain some parts of the solution.

The question is:

John and Mary are working on a job together. If John does it alone, it will take him 7 days, while Mary can do it alone in 5 days. How long will it take them to do it together?


I know that you set up \frac{1}{7}+\frac{1}{5}=\frac{1}{x}

I know that the denominators all represent total time it takes to do the job.

Thus the x value in the denominator represents the total time it takes for both John and Mary to complete the job together.

I know how to solve for x algebraically, no problem there.



The part that confuses me what the numerator is supposed to represent. For instance, is it:

\frac{1 job}{7 days}+\frac{1 job}{5 days}=\frac{1 job}{x days} or \frac{1 day worked}{7 total days}+\frac{1 day worked}{5 total days}=\frac{1 day worked}{x total days}

What "unit" or "thing" does the numerator of this rate statement represent? Are one of these guesses correct or is it something completely different?

 

[MVC5A2]

John's rate + Mary's rate = Total rate

John does 1 job per 7 days, or he works at a rate of 1job/7days.
Mary does 1 job per 5 days, or she works at a rate of 1job/5days.

So the numerator represents "1 job."

You got it right.

 

[JHamm]

Each term is the fraction of the job done each day, so John does a job in seven days which means he does \frac{1}{7}th of a job each day.

Edit: Beaten to it :)

 

[LearninDaMath]

I'm still confused by it.

In one sense, MVC, you confirm that the numerator is represented by the unit "job" so that the intended rate is \frac{job}{total days} . So this perspective is that each term is based on a total of seven days.

However, Jhamm, you explain that each term represents a fraction of the job done each day. From this perspective, what would the units of the numerator and denominator represent?

I think what I might be trying to figure out, is what Jhamm seems to already intuitively understand.

How does \frac{1job}{7totaldays} equate to \frac{1}{7} of a job per day? In the statement "\frac{1}{7} of a job per day," what units would you put next to the numerator and denominator?

 

[LearninDaMath]

Does my follow up question make sense?

 

[JHamm]

In order to complete a job in 7 days each day the person must complete a fraction of the job which when multiplied by 7 gives 1 job done. I guess the units of the fraction would be \frac{jobs}{days} so that when you multiply by the number of days you get the number of jobs completed :)

 

[spockjones20]

The two numbers, (1/7)/1, and 1/7, are the same thing. In both cases the numerators represent jobs and the denominator represents days. If you multiply (1/7)/1 by 7/7 you come up with (7/7)/7, or 1/7. I'm not sure if this is explaining things or just restating what has been posted, hopefully you gain some insight from this. Sorry if I didn't explain things the best, still a noob on this.

 

[spockjones20]

I guess I probably should have explained what those two numbers meant. The number (1/7)/1 represents how much of the job is done each day, while the number 1/7 represents how much of the job is done in seven days. I think this is how you go about this, unless I completely misunderstood something, in which case I apologize.

 

[Curious3141]

This is not a homework question. It is a question my sister asked me, and although I could solve it, I could not explain some parts of the solution.

The question is:

John and Mary are working on a job together. If John does it alone, it will take him 7 days, while Mary can do it alone in 5 days. How long will it take them to do it together?I know that you set up \frac{1}{7}+\frac{1}{5}=\frac{1}{x}

I know that the denominators all represent total time it takes to do the job.

Thus the x value in the denominator represents the total time it takes for both John and Mary to complete the job together.

I know how to solve for x algebraically, no problem there.
The part that confuses me what the numerator is supposed to represent. For instance, is it:

\frac{1 job}{7 days}+\frac{1 job}{5 days}=\frac{1 job}{x days} or \frac{1 day worked}{7 total days}+\frac{1 day worked}{5 total days}=\frac{1 day worked}{x total days}

What "unit" or "thing" does the numerator of this rate statement represent? Are one of these guesses correct or is it something completely different?

Think about it like this: John can complete 1/7 (one seventh) of the job in a day. Mary can complete 1/5 (one fifth) of the job in a day. If they work together (assuming they don't get in each other's way!),they will complete a total fraction of (1/7 + 1/5 = 12/35) of the job in a single day.

So the entire job will take 35/12 days = a little under 3 days.

The units in question are essentially jobs/day. But it can be confusing, so I recommend looking at it in the simplistic way I detailed.

 

[Joffan]

Let's solve it in a wordy sort of way.

John is a 1 job per 7 days person, while Mary is a 1 job per 5 days person. What kind of person would be equivalent to both of them working together?

John can be reckoned a 5 jobs in 35 days person with Mary a 7 jobs in 35 days person. So combining them will give us a 12 jobs in 35 days person, or a 1 job in \frac{35}{12} days person.

Of course combining people in the real world gives the opportunity for both better and worse results than this. There are jobs that are slow for anyone person alone but go immensely faster with two, and there are jobs where one person is all that can work on it and a second person should go do something else, or just make coffee. So the real answer is probably between 0.5 and 7 (the latter assuming that some other job is a better use of Mary's time).

 

[LearninDaMath]

The things that the numerator and denominator of each fraction are representing makes much more sense now. No longer does it seem like an abstract formula that just mysteriously happen to produce a corrent answer. Thanks! And haha i agree that in a real life situation, this one equation alone is probably not the perfect metric for calculating the time it would take combined individuals to complete a job - too many variables in consider indeed. Again, appreciate the feedback on this word problem.