SOLUTION: Patrice, by himself, can paint four rooms in 10 hours. If he hires April to help, they can do the same job together in 6 hours. If he lets April work alone, how long will it take h

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Patrice, by himself, can paint four rooms in 10 hours. If he hires April to help, they can do the same job together in 6 hours. If he lets April work alone, how long will it take h      Log On


   

Question 1208959: Patrice, by himself, can paint four rooms in 10 hours. If he hires April to help, they can do the same job together in 6 hours. If he lets April work alone, how long will it take her to paint four rooms?
Let A = length of time it will take April to paint four rooms

Here is my equation:
(1/10) + (1/A) = 1/6
Is this correct?


Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52792) About Me  (Show Source):
You can put this solution on YOUR website!
.

This setup is correct.

Here "1" in the setup equation represents one whole job, which is painting 4 rooms.

So, your equation (= your setup) is correct. Please accept my congratulations.

But it is not whole solution to the problem.

To complete the solution, you should find A from the setup equation.


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Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Your setup is fine, using the method shown in most references for solving "working together" problems. Patrice alone can do the job in 10 hours, so the fraction of the job he does in 1 hour is 1/10. If A is the number of hours it takes April alone to do the job, then the fraction she does in 1 hour is 1/A. And since it takes the two of them 6 hours to do the job together, the fraction they do together in 1 hour is 1/6.

Then, adding the fractions of the job each of them does alone in 1 hour gives you the fraction of the job they do together in 1 hour:

1%2F10%2B1%2FA=1%2F6

To solve the equation, multiply everything by the least common denominator, 30A.

3A%2B30=5A
2A=30
A=15

The number of hours it takes April alone to do the job is 15.

ANSWER: 15 hours

While you might be expected to set up and solve the problem by this method, there is an alternative method that many students find easier.

Consider the least common multiple of the two given times, which is 30 hours.

In 30 hours, Patrice could do the job 30/10 = 3 times.

Working together, the two of them could do the job 30/6 = 5 times.

That means that April alone can do the job 5-3 = 2 times.

And that means the number of hours it takes her to do the job alone is 30/2 = 15.

ANSWER (again, of course): 15 hours

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

You have the correct set up.
To solve you can multiply everything by the LCD 30A and you should get 3A+30 = 5A.
Which at this point should be fairly easy to solve.
The result is A = 15.

Another option is to solve like so
1%2F10%2B1%2FA+=+1%2F6

A%2F%2810A%29%2B10%2F%2810A%29+=+1%2F6 Rewrite each fraction in terms of a common denominator

%28A%2B10%29%2F%2810A%29+=+1%2F6

6%28A%2B10%29+=+10A Cross multiply

I'll let the student take over from here.

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Another approach.

Let's say the job is to paint 600 square feet of wall and ceiling area.
I arrived at this number by multiplying the given values 10 and 6.
Then tacking on a 0 at the end to make the area a bit more realistic.

This number isn't particularly important since it can be changed to anything else. You will get the same result at the end.
Also, the "four rooms" can be changed to any number of rooms to get the same answer. So we'll ignore this figure.

Patrice can paint all 600 sq ft in 10 hours when working alone.
His rate is 600/10 = 60 sq ft per hour.
rate = amountDone/time

When Patrice and April work together, assuming neither person hinders the other, they paint all 600 sq ft in 6 hours.
rate = 600/6 = 100 sq ft per hour.

April's unit rate must be 100-60 = 40 sq ft per hour since the two people's individual rates add to the combined rate 100 sq ft per hour.

Let's see how long it takes her to do the job by herself.
rate*time = amountDone
time = amountDone/rate
time = (600 sq ft)/(40 sq ft per hr)
time = (600/40) hours
time = 15 hours