SOLUTION: How do I solve this problem? Kamil and Anita are typing a manuscript. Anita can type it in 2 hours less time than Kamil. Together they can do the job in 6 hours. How long wil

Algebra ->  Rate-of-work-word-problems -> SOLUTION: How do I solve this problem? Kamil and Anita are typing a manuscript. Anita can type it in 2 hours less time than Kamil. Together they can do the job in 6 hours. How long wil      Log On


   

Question 3262: How do I solve this problem?
Kamil and Anita are typing a manuscript. Anita can type it in 2 hours less time than Kamil. Together they can do the job in 6 hours. How long will it take Anita if she works alone? Give your answer to the nearest tenth.
Answer by drglass(89) About Me  (Show Source):
You can put this solution on YOUR website!
To solve this problem, you need to identify two things: the form of the answer and the facts you have to solve the problem.

Form of the answer:
The question is asking for the time it would take Anita to type a documnet by herself so the form of the answer should be:
It would take Anita _____ hours to type the document.
Facts:
1) Anita takes 2 hours less than Kamil

2) Anita and Kamil can do the job in 6 hours

Next, we write the facts in a mathematical way:
Let's let the the letter A mean "the speed at which Anita can type the document" and let K mean "the speed at which Kamil can type the documents." From the facts, we know from the facts that it takes Anita 2 hours less than Kamil to complete the document, so let t be the time it takes Kamil to to the document, that is t*K = 1 document and (t-2)*A = 1 document. We also know that together, they can complete 1 document in 6 hours, so 6(A+K) = 1 document.

Let's collect the mathematical version of the facts:


  1. K*t = 1; Kamil can complete the document in t hours

  2. A*(t - 2) = 1; Anita can complete the document in t-2 hours

  3. (A+K)*6 = 1; Anita and Kamil can complete the document in 6 hours.



Now let's try to make some sense of these facts. First, let's rewrite the first two facts to get A and K in terms of t:


  1. K+=+%281%2Ft%29 divide both sides of equation 1 by t

  2. A+=+%281%2F%28t+-+2%29%29 divide both sides of equation 2 by (t-2)


Now we can replace A and K in the third equation to find

%28%281%2F%28t+-+2%29%29%2B%281%2Ft%29%29%2A6+=+1

To solve this equation, we do some algebra

%286%2F%28t+-+2%29%29%2B%286%2Ft%29+=+1; multiply through the equation

%286%2F%28t+-+2%29%29%28t%2Ft%29%2B%286%2Ft%29%28%28t-2%29%2F%28t-2%29%29+=+1; put the fractions under a common denominator

%28%286t%2B6%28t-2%29%29%2F%28t%28t-2%29%29%29+=+1; add the fractions together

%286t%2B6t-12%29%2F%28t%28t-2%29%29+=+1;

12t-12+=+t%28t-2%29;multiply both sides by the denominator

12t-12+=+t%5E2-2t;

Now regroup the equation to form a quadratic equation:

t%5E2-2t-%2812t-12%29+=+0

t%5E2-14t%2B12+=+0

The solution of this equation is found using the quadratic equation:
14+%2B-+sqrt%28+%28-14%29%5E2-4%2A%2812%29%29%2F2+

The solutions work out to be 13.1 and 0.9. These numbers are solutions for t, the time it takes Kamil to solve the problem. Given that it takes 6 hours for both of them to finsh the document and that Anita takes 2 hours less than Kamil, 0.9 makes no sense, so we can ignore it as a possiblity. This leaves us with Kamil taking 13.1 hours and Anita taking 11.1 hours.

Let's check these answers:
We know that (A+K)*6 = 1 (see # 3 above). A is the rate at which Anita completes the document and K is the rate at which Kamil completes the document. Therefore, %28A%2BK%29%2A6+=+%28%281%2F11.1%29+%2B+%281%2F13.1%29%29%2A6+=+.999. Since we rounded the numbers, this is a pretty good answer.