Painting a Fence - The Next Job
Chris and Zaux have decided to take on another fence paining job, as they did so well on the last job. They realise that they need some help, so they recruit Knightmare. How long would it take them to finish the job if Zaux can do it alone in 6 hours more, Chris 2 hours more and Knightmare in twice the time. I'm paying by the hour... minimum wage, mind...
- Karl Sharman
- Karl Sharman
Labels: SharedPuzzle
posted by Rajesh Lal at
10:07 PM
22 Comments:
Fence painting:
Zaux paint rate = 1/6 fence/hr.
Chris paint rate = 1/8 fence /hr.
Knightmare paint rate = 1/12 fence/hr.
In 1 hour, together they paint:
1/6 + 1/8 + 1/12 of the fence =
4/24 + 3/24 + 2/24 = 9/24 fence
Thus in 2 6/24 hrs. together they finish painting the fence
Answer - 2 1/4 hrs.
Zaux - that would be too easy.... 6 hours more, 2 hours more and twice the time...
I think they'll knock this job over in 96 mins.
Oh yea ...I missed the word "more" after Zaux's name
would take them 2 2/3 hours
nah .... 2 2/3 hrs is wrong
got 2 hours, but don't feel good about it... heh heh
Leon ... 3rd time I went through it, I too got 96 minutes. Either we are right or both made the same mistake.
Hi Karl. "More" than what? The previous 2 hours or more than the previous personal times? Is Knightmare's "twice the time" referring to the original time (2 hours) or the new time that Zaux and Chris would have achieved working together at their new slower rate?
i.e. I find the question has many interpretations.
Chris ... sounds like the problem is saying it takes x hours more when painting alone than when painting as a group.
Chris, solve this thing ... drove me crazy ... heh heh.
Thanks Zaux. That interpretation makes a lot of sense. I'll give it a go.
Time = 2(Sqrt(13)-2)/3 = 1.07036... hours.
Let Z be the time that Zaux takes to do the job acting alone, similarly C and K. Let T be the time to do the job acting together.
Then 1/T = 1/Z +1/C + 1/K
= 1/(T+6) + 1/(T+2) + 1/(2T)
Ask wolframalpha to solve this:
http://www.wolframalpha.com/input/?i=Solve%5B1%2F%28T+%2B+6%29+%2B+1%2F%28T+%2B+2%29+%2B+1%2F%282+T%29+%3D%3D+1%2FT%2C+T%5D
and you're done.
On the other hand, Karl might prefer the alternative answer, of
-3.737... hours, so we have to pay him. I bet he wishes we weren't on minimum wages now.
I'll do the boring details now.
Have 1/T = 1/(T+6) + 1/(T+2) + 1/(2T)
Mult throughout by 2T(T+6)(T+2) =>
2(T+6)(T+2) = 2T(T+2) + 2T(T+6) + (T+6)(T+2)
2T²+16T+24 = 2T²+4T + 2T²+12T + T²+8T+12
2T²+16T+24 = 5T² + 24T + 12
rearranging =>
3T² + 8T - 12 = 0
So T = (-8 +/- Sqrt(64 + 4*3*12))/6
T = (-8 +/-Sqrt(208))/6
T = 2(-2 +/- Sqrt(13))/3
That's far enough.
So when painting alone, Zaux takes just over 7 hours, Chris just over 3 hours, and Knightmare just over 2 hours. So I suspect that the given data need serious scrutiny.
This post has been removed by the author.
Hi Cam. Posts crossed.
I haven't much experience of wolframalpha. But if you follow the link I gave, you'll have an idea of how to ask it to solve a problem.
Hi Cam, just caught you before posting this. I didn't understand the question until Zaux gave me his interpretation. I don't know where he was going wrong in his calculations. You might like a couple of problems I posted back in August/September. I'm sure you'll also like the "Bug on a Band" 25th Aug and possibly "Little 2" 16 July.
Hmmm ... encouraging ... with my meager math skills, I actually got to:
3T^2 + 8T - 12= 0
and then didn't solve it because I thought it was wrong ... heh heh.
Hi Zaux. Ah, that's a relief, I thought ou might have got the basic moves wrong. I'm not sure if it's agood or bad thing that you were skeptical about you result.
Anyway, I'll use this as an excuse to derive the solution to the quadratic equation. It's frequently needed, but it's seems to be taken for granted that everyone knows it:
Let ax² + bx + c = 0, then
x² + (b/a)x + c/a = 0.
Consider (x + (b/a)/2)² = x² + (b/a)x + (b/a)²/4.
Substituting into the previous equation =>
(x + (b/a)/2)² + (c/a) - (b/a)²/4 = 0
=> (x + (b/a)/2)² = (b/a)²/4 - (c/a)
= (b² - 4ac)/(2a)²
sqrting => x + (b/a)/2 = +/-Sqrt((b² - 4ac)/(2a)²)
blah, blah
x = (-b +/-Sqrt(b² - 4ac)/(2a)
OK school's over.
Sorry folks... been busy at work.
Zaux' interpretation is the same as mine - "sounds like the problem is saying it takes x hours more when painting alone than when painting as a group".
The source comes back with an answer of 40 minutes, which I could not validate. That's why I posted it here! Probably wrong to do so without a working answer.
Hi Zaux. I should have added, that it's often the case that the solution to a problem involves a quadratic (or higher) equation. The trick is to decide which of the two (or more answers) is the most likely one. Fortunately, one of the answers is usually the obvious one. e.g. you can't finish painting a fence before you began without the aid of a TARDIS (or Pequod I expect).
Sadly, I can't cite an example, but once in a while, the "wrong" answer turns out to be the most interesting one.
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