SOLUTION: It takes Robert 9 hours longer to construct a fence than it takes Eldin. If they work together, they can construct the fence in 20 hours. How long would it take each, working alon

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Question 1170604: It takes Robert 9 hours longer to construct a fence than it takes Eldin. If they work together,
they can construct the fence in 20 hours. How long would it take each, working alone,
to construct the fence?
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

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+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 or x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-31%29%2Bsqrt%28%28-31%29%5E2-4%281%29%28-180%29%29%29%2F%282%281%29%29 or x+=+%28-%28-31%29-sqrt%28%28-31%29%5E2-4%281%29%28-180%29%29%29%2F%282%281%29%29

x+=+%2831%2Bsqrt%281681%29%29%2F%282%29 or x+=+%2831-sqrt%281681%29%29%2F%282%29

x+=+%2831%2B41%29%2F%282%29 or x+=+%2831-41%29%2F%282%29

x+=+%2872%29%2F%282%29 or x+=+%28-10%29%2F%282%29

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

---------------------------------------------------------

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

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

It takes Robert 9 hours longer to construct a fence than it takes Eldin. If they work together, they can construct the fence in 20 hours. How long would it take each, working alone, to construct the fence?
Let time it takes Robert to do the job, be R

Then time it takes Eldin is: R - 9

Therefore, Robert can do 1%2FR of job in 1 hour, while Eldin can do 1%2F%28R+-+9%29 of job in 1 hour

We then get the following equation: matrix%281%2C3%2C+1%2FR+%2B+1%2F%28R+-+9%29%2C+%22=%22%2C+1%2F20%29

20(R - 9) + 20R = R(R - 9) ------- Multiplying by LCD, 20R(R - 9)



(R - 45)(R - 4) = 0

Robert's time to complete job, alone: highlight_green%28matrix%281%2C2%2C+45%2C+hours%29%29                OR                 R = 4 (ignore)

Eldin's time to complete job, alone: highlight_green%28matrix%281%2C4%2C+45+-+9%2C+%22=%22%2C+36%2C+hours%29%29