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        riddles >> medium >> painting the wall
        
(Message started by: antkor on Dec 23^rd, 2013, 9:40am)

Title: painting the wall
Post by antkor on Dec 23^rd, 2013, 9:40am

Three painters A, B and C are painting a wall. If painter A was painting alone, then he would need 5 hours more to paint the wall than all of them combined. If B was painting alone he would need 8 hours more than all of them combined and if C was painting alone he would need 1 hour more than the 3 of them combined. How much time do A and C together to paint the wall? Note: the rythm they paint remains constant.

Title: Re: painting the wall
Post by rloginunix on Dec 24^th, 2013, 3:01am

I've got [hide]1 hour 48 minutes 20 seconds[/hide].

Title: Re: painting the wall
Post by antkor on Dec 24^th, 2013, 6:06am

on 12/24/13 at 03:01:03, rloginunix wrote:

I've got [hide]1 hour 48 minutes 20 seconds[/hide].

That is correct, but since you posted the solution, would you mind sharing your analysis with the rest of the members? for example the equations you used etc.

Title: Re: painting the wall
Post by towr on Dec 24^th, 2013, 10:24am

He just want to leave something for other members to do. :)

[hide]
1/A = 1/(A+B+C) + 5 = 1/C + 4
1/B = 1/(A+B+C) + 8 = 1/C + 7
1/C = 1/(A+B+C) + 1

To determine: 1/(A+C)

Replace A, B
1/C = 1/(1/(1/C + 4)+1/(1/C + 7)+C) + 1
Simplify
14 C³ + 11 C² - 4C -1 = 0
Factor
(C+1)*(14C² - 3C -1) = 0
We may safely assume C > 0, so
(14C² - 3C -1) = 0

C ~= 0.39608 h
1/A = 1/C + 4 ~= 6.52474, A ~= 0.15326
1/(A+C) ~= 1.82036 h ~= 1:49:13[/hide]
There's some rounding differences. [hide]The exact value is (3*sqrt(65) + 5)/16, which translates to 1:49:27 [/hide]

Actually, the puzzle's been asked before here (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1349427497;).

Title: Re: painting the wall
Post by antkor on Dec 24^th, 2013, 11:06am

ah didn't know the riddle has been already posted. my apologies... you can move it there or delete it since it has no reason to exist.

Title: Re: painting the wall
Post by rloginunix on Dec 24^th, 2013, 2:27pm

No, no. I'm here. Had some pre-Christmas running around to do. Sorry for taking long time to respond.

I also didn't know that this puzzle already existed on the forum. My apologies.

My solution path was similar to towr's. I just took a slightly more generic first step and made a few observations along the way.

[hide]
Let T be the total number of hours it takes the entire team to paint the entire wall. Let A, B and C be the number of hours it takes painters A, B and C to paint the entire wall if working alone. Then if all three painters work on the wall at the same time painter A will paint a portion of the wall expressed as T/A. For painter B it will be T/B. And for painter C it will be T/C.

The whole wall is 1 and it's a sum of portions painted by all the participants:

T/A + T/B + T/C = 1

0r

1/A + 1/B + 1/C = 1/T

Now this equation can be used in various similar problems. In this particular case the variables A, B, C and T can be animated in 4 different ways - by making only one entity an unknown and expressing the rest of them via it.

For example. Let's do it via painter A. Let X be the numb er of hours it takes painter A to cover the entire wall. Then according to the first condition the entire team will take X - 5 hours for the entire job. Painter B will take X - 5 + 8 = X + 3 hours. Painter C will take X - 5 + 1 = X - 4 hours. In our notation:

A = X
B = X + 3
C = X - 1
T = X - 5

The equation above then becomes:

1/X + 1/(X + 3) + 1/(X - 1) = 1/(X - 5)

Solve it for X and the rest of the unknowns follow. But you can go through the same exercise and frame it in terms of say T. Meaning let it be the unknown. Then the equation becomes:

1/(X + 5) +1/(X + 8.0) + 1/(X + 1) = 1/X

and so on. (8.0 is to get rid of the smiley). All in all you'll get 4 different cubic equations but the answer of course should be only one.

For our original case we get the equation:

X^3 - 8X^2 + 5X +30 = 0

or

(X - 3)(X^2 - 5X - 10) = 0

The first root 3 is no good - it must be more than 5. The second root is negative. The third is positive and X = A = 6.533.

From that B = 9.5 and C = 2.5. Now we can answer the question of the problem:

1/A + 1/C = 1/T

or

T = AB/(A + B)

In this case I've got 6.5*2.5/9.0 = 1.80555 and as towr pointed out here are the rounding differences but the idea is the same.

A bit more generic then direct towr's solution but I hope it adds a slightly different perspective.
[/hide]

Title: Re: painting the wall
Post by rloginunix on Dec 24^th, 2013, 8:04pm

By now this forum is very mature. And it feels that the following theorem holds true:

the number of puzzles/riddles/problems that the moderators do not know about tends to zero.

I'll give it a shot though to entertain the audience with the following classic - Newton's Cows. I've searched all the reasonable areas of this forum for the pattern "Newton cow" and the search came up blank. So this is just a link up to the next thread. Look for "Newton's Cows" here in medium.

Title: Re: painting the wall
Post by rloginunix on Dec 24^th, 2013, 8:38pm

Sorry. It's me again.

I now looked at the link towr provided. My solution path is indeed almost identical but, for what it's worth, in my path you'll find an extra observation - that you can animate the variables in four different ways and frame four different cubic equations. So if the moderators are willing may be it makes sense to merge the two threads into one for a full picture. Just an idea.