Question 3262
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<br>
2) Anita and Kamil can do the job in 6 hours<br>

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.<br>
Let's collect the mathematical version of the facts:<br>
<ol>
<li>K*t = 1; Kamil can complete the document in t hours</li>
<li>A*(t - 2) = 1; Anita can complete the document in t-2 hours</li>
<li>(A+K)*6 = 1; Anita and Kamil can complete the document in 6 hours.</li>
</ol>
<hr>
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:<br>
<ol>
<li>{{{K = (1/t)}}} divide both sides of equation 1 by t</li>
<li>{{{A = (1/(t - 2))}}} divide both sides of equation 2 by (t-2)</li>
</ol>
Now we can replace A and K in the third equation to find<br>
{{{((1/(t - 2))+(1/t))*6 = 1}}}<br>
To solve this equation, we do some algebra<br>
{{{(6/(t - 2))+(6/t) = 1}}}; multiply through the equation<br>
{{{(6/(t - 2))(t/t)+(6/t)((t-2)/(t-2)) = 1}}}; put the fractions under a common denominator<br>
{{{((6t+6(t-2))/(t(t-2))) = 1}}}; add the fractions together<br>
{{{(6t+6t-12)/(t(t-2)) = 1}}}; <br>
{{{12t-12 = t(t-2)}}};multiply both sides by the denominator <br>
{{{12t-12 = t^2-2t}}}; <br>
Now regroup the equation to form a quadratic equation:<br>
{{{t^2-2t-(12t-12) = 0}}}<br>
{{{t^2-14t+12 = 0}}}<br>
The solution of this equation is found using the quadratic equation:
{{{14 +- sqrt( (-14)^2-4*(12))/2 }}}<br>
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.
<hr>
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, {{{(A+K)*6 = ((1/11.1) + (1/13.1))*6 = .999}}}. Since we rounded the numbers, this is a pretty good answer.