Question 192678
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Let <b><i>x</i></b> represent the number of hours that Tammy would take, then <b>2<i>x</i></b> is the number of hours that Jeri would take, and <b>2<i>x</i> - 2</b> is the number of hours that Laura would take.


Since they took 1 hour and 20 minutes *[tex \large \left(\frac{4}{3}\ \text{hour}\right)], they can do *[tex \large \left(\frac{3}{4}\right)] of the job in 1 hour.


Tammy can do *[tex \large \left(\frac{1}{x}\right)] of the job in 1 hour.


Jeri can do *[tex \large \left(\frac{1}{2x}\right)] of the job in 1 hour.


Laura can do *[tex \large \left(\frac{1}{2x - 2}\right)] of the job in 1 hour.


So, together they can do:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{x} + \frac{1}{2x} + \frac{1}{2x - 2} = \frac{3}{4}]


The LCD is *[tex \large 2x(2x-2) = 4x^2 - 4x], so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{4x - 4}{4x^2 - 4x} + \frac{2x - 2}{4x^2 - 4x} + \frac{2x}{4x^2 - 4x} = \frac{3(x^2 - x)}{4x^2 - 4x}]


Combining like terms:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{3x^2 - 11x + 6}{4x^2 - 4x} = 0]


Multiply by *[tex \large 4x^2 - 4x], making a note to exclude the values of 0 and 1 from the ultimate solution set because these values would make the denominator = zero.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x^2 - 11x + 6 = 0]


Which factors:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (3x - 2)(x - 3) = 0]


Meaning that Tammy would take either *[tex \large \frac{2}{3}] of an hour or 3 hours to do the job.  The *[tex \large \frac{2}{3}] answer is an absurdity because that would mean that Jeri could do the job in *[tex \large \frac{4}{3}] which is the time that it took all three to actually do it.  Exclude *[tex \large \frac{2}{3}] as an extraneous root.  The correct answer is then 3 hours for Tammy, two times that, or 6 hours for Jeri, and two hours less than that, or 4 hours for Laura.


Check:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{3} + \frac{1}{6} + \frac{1}{4} = \frac{4}{12} + \frac{2}{12} + \frac{3}{12} = \frac{9}{12} = \frac{3}{4}].


So they can do three-fourths of the job in 1 hour, or the whole job in the reciprocal of that, or four-thirds hour which is 1 hour and 20 minutes.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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