Re: Please help - comments much appreciated...

almurph@xxxxxxxxxxxxx wrote:
> Hi everyone,
>
> Can you help me please? I've always had problems with puzzles of the
> following nature:
>
> If a "certain amount of machines" make a "certain amount of widgets" in
> a "certain amount of time" then how long does it take for x amount of
> machines to make y amount of widgets...?
>
>
> For example:
>
> It takes 5 machines 5 minutes to make 5 widgets.
> How long does it take 100 machines to make 100 widgets?
>
>
> Another example along this theme is:
>
> If 1 and a half chickens take 1 and a half days to make 1 and a half
> eggs how long does it take for 9 chickens to make 9 eggs?
>

Well, You probably find out by thinking about it long enough using
common sense, but here is the systematic approach:

+---------------------------+
|Look out for the invariant!|
+---------------------------+

You are given some circumstances and are asked for the outcome under
other circumstances.
The trick is to see what does *not* change, i.e. find the invariant.
In these cases the invariant is the "productivity" of the machine:
Assume that a machine produces x widgets per minute, i.e the production
speed
is x widgets per minute per machine.
=> 5 machines produce 5*x widgets per minute, thus 5*5*x widgets in 5
minutes
Thus 5*5*x = 5, or x=1/5.
Next, 100 machines of the same productivity (invariant!) produce 100*x
widgets per minute, i.e. 20 widgets per minute, thus the time needed is
100 widgets / (20 widgets/minute) = 5 minutes.
Note: It is helpful to use units (e.g. widgets per minute per machine)
in order to be forced to make the right choice between multiplying and
dividing ;)

The second problem uses half chickens and half eggs. That's funny, but
shouldn't disturb us.
Apparently the invariant here is
1.5 eggs per 1.5 days per 1.5 chickens = 2/3 egg/(day*chicken)
Then
2/3 egg/(day*chicken) * 9 chickens = 6 eggs per day,
i.e. we need to wait
9 eggs / (6 eggs/day) = 1.5 days
(In general, if the number of eggs needed equals the number of
available chickens, we obtain 1.5 days).
One should be a bit careful here. For example, is it true that 129600
chickens need 1 second to produce 1 egg? If the chickens are in random
phases of the production cycle, the answer is "yes, on the average". On
the other hand, if the chickens are "in sync", You wait a day and a
half for You first egg, but then You get lots of them at once...

Here's another riddle of that kind:
A marching band of 5 musicians plays a march in 3 minutes.
How long does a band of 10 musicians need for the same march?
;)

hagman

.

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