SOLUTION: Thomas can rake the leaves in his backyard in 4 hours. Working together Thomas and his sister Joanna can rake the leaves in 2.5 hours. How many hours would it take Joanna to rake

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Question 1091218: Thomas can rake the leaves in his backyard in 4 hours. Working together Thomas and his sister Joanna
can rake the leaves in 2.5 hours. How many hours would it take Joanna to rake the leaves by herself?

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52890)   (Show Source):
You can put this solution on YOUR website!
.

Their combined rate of work is = of the job per hour. // Notice that = hours = 2.5 hours. Thomas' individual rate of work is of the job per hour. Hence, Joanna's rate of work is = = of the job per hour. It means that Joamnna will complete the job in = hours = 6 hours and 40 minutes working alone.

Solved.

It is a typical joint work problem.

There is a wide variety of similar solved joint-work problems with detailed explanations in this site. See 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.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic
"Rate of work and joint work problems" of the section "Word problems".

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Some people use equation/equations to solve such problems.

But actually, these problems can be solved in much simpler way, without using equations, as I did in my solution.

Not only it is simpler - it helps students to understand better the meaning of things behind the solution.

So, manipulating with the fractions only is ALL what you need to get the solution in this case.

But for it, you need to understand well what you are doing.

My principle is

"The solution must be as simple as the problem requires. It should not be more complicated, unless special circumstances force you to do it in other way."

Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!

The traditional way to work a problem like this is to think in terms of the fraction of the job completed by each worker in an hour. Since Thomas can do the job in 4 hours, he does 1/4 of the job in 1 hour.

We don't know how many hours Joanna takes to do the job herself. That is what we are to find in the problem; so call it x. Then the fraction of the job she does in 1 hour is 1/x.

So working together for an hour, the fraction of the job they can do together is

But we know it takes them 2.5 (or 5/2) hours to do the job together; so

Multiplying by the least common denominator 20x...

It takes Joanna 20/3 hours, or 6 2/3 hours, or 6 hours and 40 minutes, to do the job by herself.

And here is another method that I find many student like better, because it avoids working with fractions. The arithmetic involved in this solution method is very similar to some of the work we did above using the traditional solution method.

For this method, we find a convenient common multiple of the times given in the problem. In your example, we know one worker takes 4 hours to do the job alone, and we know the two together can do the job in 2.5 hours.

A convenient common multiple of 4 and 2.5 is 20. So consider what the workers could do in 20 hours.

Thomas can do the job in 4 hours; so in 20 hours he could do the job 5 times.

The two together can do the job in 2.5 hours; so in 20 hours they could do the job 8 times (20 divided by 2.5).

But if together they can do the job 8 times in 20 hours, and Thomas alone can do the job 5 times in 20 hours, then Joanna alone can do the job 3 times in 20 hours.

And so the number of hours it takes Joanna alone to do the job one time is 20/3 hours, or 6 hours and 40 minutes.