SOLUTION: two workmen can paint a house together in 6 and 1/3 days. in how many days can each paint it alone if it takes one of them two days longer than the other?

Question 903411: two workmen can paint a house together in 6 and 1/3 days. in how many days can each paint it alone if it takes one of them two days longer than the other?
Found 2 solutions by stanbon, richwmiller:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
two workmen can paint a house together in 6 and 1/3 days. in how many days can each paint it alone if it takes one of them two days longer than the other?
-----------------------
Together DATA::
time = 19/3 days/job ; rate = 3/19 job/day
---
One's DATA:
time = x days/job ; rate = 1/x job/day
--------
Other's DATA:
time = x+2 day/job ; rate = 1/(x+2) job/day
-------------------------
Equation:
rate + rate = together rate
1/x + 1/(x+2) = 3/19
19(x+2) + 19x = 3x(x+2)
-----
38x + 38 = 3x^2 + 6x
-----
3x^2 - 35x - 38 = 0
(3x-38)(x+1) = 0
Positive solution::
x = 38/3 = 12 1/3 days (time for One to do the job alone)
x+2 = 14 1/3 days (time for Other to do the job alone)
-----------------
Cheers,
Stan H.
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Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
1/x+1/(x+2)=1/6.333
(6.333*x*(x+2))/x+(6.333*x*(x+2))/(x+2)=(6.333*x*(x+2))/6.333
(6.333*(x+2))+6.333*x=x*(x+2)
6.333*x+12.666+6.333*x=x^2+2*x
12.666*x+12.666=x^2+2*x
x^2+-10.666*x-12.666=0

Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics

Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert to standard form by dividing both sides by 1:
We have: . What we want to do now is to change this equation to a complete square . How can we find out values of somenumber and othernumber that would make it work?
Look at : . Since the coefficient in our equation that goes in front of x is -10.666, we know that -10.666=2*somenumber, or . So, we know that our equation can be rewritten as , and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that is equivalent to our original equation .

The highlighted red part must be equal to -12.666 (highlighted green part).

, or .
So, the equation converts to , or .

Our equation converted to a square , equated to a number (41.106889).

Since the right part 41.106889 is greater than zero, there are two solutions:

, or

Answer: x=11.7444654331128, -1.07846543311278.

11.7444654331128 days
13.7444654331128 days
check
a=11.7444654 days
b=13.7444654 days
1/a+1/b=1/x
1/11.7444654+1/13.7444654=1/x
1/11.7444654+1/13.7444654=1/6.333
1/x=1/6.333
x=6.333 days working together

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