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

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) (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:

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

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

(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:

divide both sides of equation 1 by t

divide both sides of equation 2 by (t-2)

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

To solve this equation, we do some algebra

; multiply through the equation

; put the fractions under a common denominator

; add the fractions together

;

;multiply both sides by the denominator

;

Now regroup the equation to form a quadratic equation:

The solution of this equation is found using the quadratic equation:

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, . Since we rounded the numbers, this is a pretty good answer.
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