Working Rates Problem 2 The boat hit something, which caused a hole in the boat. Water is leaking in at a consistent rate and some has already gathered when the hole is detected. At this stage, twelve men of the same skill can make the boat dry in 3 hours, while five men require ten hours. How many men are required to make the boat dry in 2 hours and how would you find the answer to this?
I thought that it would require 13 people to do it in 2 hours as if you increase the number of people by 7, the time decreases by 7 hours. So I thought increasing the number of people by 1 would decrease the time by 1. So what I did was add 1 to 12, which would decrease the time (3 hours) by 1 hour. I don't think this is the correct way to do it though...
 A: No, that is not good reasoning. Forget about the leak for now, and think about this. Suppose it takes 2 hours for 2 men to stuff 100 envelopes. Then I think it will be clear that it will take 1 man twice as long, i.e. 4 hours to stuff 100 envelopes ... So with your reasoning, it will take 6 hours for 0 men to stuff 100 envelopes?! Or, for that matter, 3 men will stuff all 100 envelopes in 0 hours?  And 4 men will do it in -2 hours?!
Clearly the math doesn't work out linearly like that!
OK, try this:
Let $z$ be how much water there is when they find the leak (in some kind of unit of measurement, gallons, liters, it doesn't really matter)
Let $x$ be how much water goes in per hour (same units)
let $y$ be how much water 1 man is able to remove per hour (same units)
Then we are told:
$z + 3x - 3 *12y=0$
and
$z+ 10x - 10*5y=0$
From this, you can figure out $z$ and $x$ in terms of $y$, and then you need to solve:
$z + 2x - 2*N*y=0$
with $n$ the number of men needed to finish the job in 2 hours.
Good luck!
If you still can't find it yourself, mouse over the below:

 Subtracting the first equation from the second, we get $7x=14y$, and thus $x=2y$


 Filling that back into the first equation we get $z+6y-36y=0$ and thus $z=30y$

 Filling $x=2y$ and $z=30y$ into third equation we get $30y+4y-2Ny=0$, and thus $N=17$

A: Bram28 has explained why your line of reasoning doesn't work.
As for the approach, it's best to create a few variables and express the problem in terms of them using equations for this kind of question.

Let the initial amount of water in the boat when the hole is detected be $a$.
Let the constant rate of flow of water into the boat be $b$.
Let the (constant) rate at which 1 man empties the boat of water be $c$.
Then the amount of water in the boat at any given time, $t \ge 0$ with $0$ being the time the hole was detected, with $k$ men at work is 
$$W_k(t)=a + (b-kc)t$$
(assuming the number of men is proportional to the rate at which they empty the boat).

See if you can solve the problem from just this and if you get stuck, you can look below for hints.

We are given from the question:
$$W_{12}(3)=a + 3(b-12c) = 0$$
$$W_5(10)=a + 10(b-5c) = 0$$
From this, you get
$$3b - 36c = 10b - 50c$$
$$\implies b = 2c$$
Substituting back in, you get
$$a = 30c$$
Substituting again, you get
$$W_k(t)=(30 + (2-k)t)c$$
Now you are asked to find $k$ such that $W_k(2)=0$.
$$\implies (30 + 2(2-k))c=0$$
$c \neq 0$
$$\implies 30 + 4-2k=0$$
$$\implies k=17$$
