The following question was asked on a math test:

If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?

Ryan answered 12, Stefanie said 13, and Tom put down 14 for his answer. After looking at their work, the teacher decided to give all three students credit.

What were the arguments that each student made?

If 6 cats kill 6 rats in 6 minutes...

12 - Ryan
Assumption:
The number of cats required is proportional to rats and inversely proportional to time required.

So, we need to multiply the number of rats by 100/6 and multiply the minutes by 50/6...

This means that we multiply the number of cats by 100/6 * 6/50 = 2
6x2 = 12



13 - Stefanie
Assumption:
Apparently each cat takes 6 minutes to kill one rat (which is killed at the end of each 6 minute duration).

In the initial case, we had 6 cats operating in parallel. If we have 8 times as much time (48 minutes), we will kill 8 times as many rats (8x6=48).. So, if we double the number of cats (12), we'll also double the number of rats killed (to 96)..

With only 12 cats, we'd have 96 killed at the end of 48 minutes, and 108 killed at the end of 54 minutes, but in essence we can't wait until 54 minutes, so we need another cat to get it up to at least 100 by 50 minutes. This will give us 13 x 7 = 91 killed at the end of 6x7=42 minutes, 104 killed at then end of 48 minutes, (so clearly, we've got our 100 dead at 50 minutes).



14 - Tom
Assumption:
Apparently a cat can't catch rats alone, but a pair of cats teams up on each rat and take 3 minutes to to kill it (which is killed at the end of each 3 minute duration).

In other words 3 pair kills 6 rats in 6 minutes, or:
1 pair kills 2 rats in 6 minutes, or
1 pair kills 1 rat in 3 minutes.

In the initial case, we had 3 pair operating in parallel.
If we have 6 pair operating in parallel, we will get kill 6 rats every 3 minutes.

So, at the end of 48 minutes we will have 6 x (48/3) = 96 rats killed. It's not until the 51st minute that we will have 6x(51/3) = 102 rats killed, but we need 100 killed by 50 minutes....

So, we need to add a 7th pair of cats (total of 14 cats).

Posted by SilverKnight on 2004-01-20 12:52:25