SOLUTION: Patrick, 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

Question 1207751: Patrick, 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 me see.

Patrick = 1/10

April = 1/x

Together = 1/6

(1/10) + (1/x) = (1/6)

Is this the correct equation?
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.

        Yes, this setup equation is correct.

        You can continue and solve it.

        But this problem can be solved without using any equation,
        by manipulating fractions, ONLY.

        See the reasoning below.

Patrick and April can do the job together in 6 hours. Hence, their combined rate of work is of the job per hour. Patrick, working alone, can do the job in 10 hours. Hence, Patrick's rate of work is of the job per hour. Hence, April's rate of work is the difference - = = = = . It means that April makes 1/15 of the job per hour. Hence, April needs 15 hours to complete the job working alone.

At this point, the problem is solved in full, with complete explanations.

==================

So, you can solve this problem in two ways :   manipulating with fractions or using equations.

Both methods have equal rights to exist.

Both methods implement the same idea: making a balance of rates of work.
One method implements it using manipulations with fractions;
other method implements it using equations.
But the idea of the solution is the same in both methods - only the implementation forms are different.

Solving by manipulating fractions is accessible method for 5-th or 6-th grade students.

Solving using equations is accessible method for 7-th grade students, who are just familiar with equations.

To see many other similar (and different) problems on joint work,
solved with complete explanations to teach you, look into the lessons

    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Selected joint-work word problems from the archive

Read them and get be trained in solving joint-work problems.

Consider these lessons as your textbook, handbook, guidebook, tutorials and (free of charge) home teacher,
which is always with you to help and to inspire.

/////////////////////

By the way, this problem was solved at this forum many years ago (perhaps, 20 years ago).
For the solutions, see the links

https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.question.264833.html

https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.68586.html#google_vignette

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

You have the correct set up so far.

I'll show two ways of solving that equation.
Here is one way.
1/10 + 1/x = 1/6
30x(1/10 + 1/x) = 30x(1/6)
3x + 30 = 5x
30 = 5x-3x
30 = 2x
2x = 30
x = 30/2
x = 15 hours is the amount of time April takes when working alone.

Or you can solve it like this.
1/10 + 1/x = 1/6
x/(10x) + 10/(10x) = 1/6
(x+10)/(10x) = 1/6
6(x+10) = 10x
6x+60 = 10x
60 = 10x-6x
60 = 4x
4x = 60
x = 60/4
x = 15 hours

--------------------------------------------------------------------------

Another approach

Let's say hypothetically that the total surface area to paint all four rooms is 600 square feet.

Patrick, when working alone, can paint all 600 square feet in 10 hours.
His unit rate is 600/10 = 60 sq ft per hour.
Formula: rate = amountDone/time

When Patrick and April work together, they get the job done in 6 hours.
I'll assume neither person slows down the other.
Their combined unit rate is 600/6 = 100 sq ft per hour.

Therefore, April's unit rate must be 100-60 = 40 sq ft per hour when she works alone.

Let's determine how long April will take when she does the job alone.
rate*time = amountDone
time = amountDone/rate
time = 600/40
time = 15 hours

The hypothetical 600 sq ft figure I came up with isn't special at all. Turns out you can replace it with any other positive number to get the same final answer marked in red.
I picked 600 since 10*6 = 60, and then I tacked a zero at the end to make the area appear a a bit more realistic pertaining to those four rooms.

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

Here is another way to solve this kind of problem without using fractions.

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

In 30 hours, Patrick himself could paint the four rooms 30/10 = 3 times; in 30 hours, the two together could paint the four rooms 30/6 = 5 times.

That means in 30 hours April alone could paint the four rooms 5-3 = 2 times.

And that means she could paint the four rooms alone in 30/2 = 15 hours.

ANSWER: 15 hours.

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