I admit to being slavishly devoted to Neopets. That being said, there’s a riddle up over there that boggles my mind, because it’s math, and I’m horrible at math. So, they’re building a castle, and:
84?
I actually solved this at a guess… since I didn’t think they’d involve fractional days, I started with T, the total days spent, at 84, the smallest number that would have both T/12 and T/7 come out as whole numbers. That gives us:
Foundations - T/6 - 14
Walls - T/12 - 7
Roof - T/7 - 12
Windows and doors - 5
Floors - T/2 - 42
Moat - 4
And that makes up 84 nicely.
The harder way would be to add up T/6, T/12, T/7 and T/2, make an equation where T subtract all of those equals 5 + 4, and then solve for T.
Okay, my first attempt to get this into equation form is:
x/6 + x/12 + x/7 + 5 + x/2 + 4 =x, where X=total construction time expressed in days.
Multiply everything out to get:
14x/84+7x/84+12x/84+42x/84+4=x
Add together to get:
75x/84 + 9=x
Here I hit a snag–I’m not sure why–but the answer is pretty clearly 84 days.
CHECK:
84/6=14
84/12=7
84/7=12
5=5
84/2=42
4=4
14+7+12+5+42+4=84
Daniel
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x
multiply by the common denominator of 84:
14x + 7x + 12x + 420 + 42x + 336 = 84x
75x + 756 = 84x
756 = 9x
x = 84 days
Thanks for the algebra brush-up.
From the text, the entire time, T, obeys the equation:
T = T/6 + T/12 + T/7 + 5 + T/2 + 4
multiplying both sides by 84:
84T = 14T + 7T + 12T + 420 + 42T + 336
summing:
84T = 75T + 756
subtracting 75T from both sides:
9T = 756
dividing both sides by 9:
T = 84
Basically, we know how long they spent working on the activities we don’t have a ratio for.
This is 4+5=9 days.
We can work out the ratio of time spent on these two activities together, since that must account for the ‘missing’ ratio:
(1/2) + (1/6) + (1/7) + (1/12) + ? = 1
To find the ‘missing’ ratio, we subtract the other ratios from 1.
? = 1 - (1/2) - (1/6) - (1/7) - (1/12)
We know that this missing ratio is equivalent to 9 days. Therefore, if we divide the ratio by 9, we get the ratio for one day. If we then divide 1 (the whole project) by this ratio, we get the number of days it took - 84.
This is a simplified version of Diophantus’ epitaph, with a few words changed here and there to disguise it.
Thanks, folks! I knew the Dopers could do it.
This looks like the kind of algebra they tried to teach me in 9th grade, but it bamfoozled me then just as much as it does now. The closest I can come to explaining what all those equations look like to me is to compare them to a sentence like: Run frankfurter dog cloudy for mustard. The individual words make sense, but your brain can’t make them work together. That’s the way I am with most math. I used to have a recurring nightmare that my 9th grade algebra teacher was chasing me and catching me and throwing me into a big cooking pot full of alphabet soup, only instead of the alphabet it was all numbers, and the whole time he was yelling at me, “Factoring! Factoring!”
Yeah, I probably should have gotten a tutor or something…
Most importantly, 84 = 42 x 2. And as every good Doper knows, 42 is the answer. Just not to your question.
I dunno… “What is half the number of days needed to complete the Neopets castle” does have a rather nice cosmic ring to it…
All I know is that 42 days just working on the floors seems awfully inefficient. They should find a new flooring contractor.
Or a psychiatrist. Then again, factoring is rather maniacal stuff.
x[sup]2[/sup] + 7x + 12 = (x+3)(x+4)!! Factoring! FACTORING! BWAHAHAHAHAHAH!!
And back on the Valium I go…