Question 1170604
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x = time it takes Eldin to do the job on his own
x+9 = time it takes Robert to do the job on his own
x+9 is 9 hours longer compared to x
x is some positive real number.


If it takes x hours for Eldin to do the job alone, then his rate is 1/x jobs per hour. Robert's rate is 1/(x+9) jobs per hour if he works alone.


Their combined rate is 1/x + 1/(x+9)


This is equal to the combined rate 1/20 since they can get the job done in 20 hours if they work together. One worker must not slow the other down.


The equation we must solve is
1/x + 1/(x+9) = 1/20



Clear out the fractions by multiplying everything by the LCD 20x(x+9)
Doing so yields the following
1/x + 1/(x+9) = 1/20
20x(x+9)*1/x + 20x(x+9)*1/(x+9) = 20x(x+9)*1/20
20(x+9) + 20x = x(x+9)


From here let's expand everything out and then get everything to one side
20(x+9) + 20x = x(x+9)
20x + 180 + 20x = x^2 + 9x
40x + 180 = x^2 + 9x
0 = x^2 + 9x - 40x - 180
x^2 - 31x - 180 = 0


We have something of the form 
ax^2 + bx + c = 0
With a = 1, b = -31, c = -180
Those three values can be plugged into the quadratic formula
{{{x = (-b+sqrt(b^2-4ac))/(2a)}}} or {{{x = (-b-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-31)+sqrt((-31)^2-4(1)(-180)))/(2(1))}}} or {{{x = (-(-31)-sqrt((-31)^2-4(1)(-180)))/(2(1))}}}


{{{x = (31+sqrt(1681))/(2)}}} or {{{x = (31-sqrt(1681))/(2)}}}


{{{x = (31+41)/(2)}}} or {{{x = (31-41)/(2)}}}


{{{x = (72)/(2)}}} or {{{x = (-10)/(2)}}}


{{{x = 36}}} or {{{x = -5}}}


Since x must be positive, this rules out the second solution. It makes no sense to have x be negative.


The only practical solution is x = 36


It takes Eldin exactly x = 36 hours to do the job on his own
It takes Robert exactly x+9 = 36+9 = 45 hours to do the job on his own


Checking the answers:
1/x + 1/(x+9) = 1/20
1/36 + 1/(36+9) = 1/20
1/36 + 1/45 = 1/20
5/180 + 4/180 = 1/20
9/180 = 1/20
1/20 = 1/20
The answers are confirmed


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Answers:
Eldin = 36 hours
Robert = 45 hours
Values are exact
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