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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?
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Together DATA::
time = 19/3 days/job ; rate = 3/19 job/day
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One's DATA:
time = x days/job ; rate = 1/x job/day
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Other's DATA:
time = x+2 day/job ; rate = 1/(x+2) job/day
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Equation:
rate + rate = together rate
1/x + 1/(x+2) = 3/19
19(x+2) + 19x = 3x(x+2)
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38x + 38 = 3x^2 + 6x
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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)
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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.
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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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