Understanding ratios in real world problems. I'm studying maths as an adult, and I thought everything was going well until I hit the following activity. I have the answer in my workbook, but I just don't seem to be able to come to terms with the process.
"8 pumps working for 10 minutes raise 440 litres of water. How long will it take 6 pumps to raise 396 litres."
The process I tried was thus.
Pumps decrease at a ratio 6:8
Water drawn decreases at a ratio of 396:440
So I then try 10 * 6/8 * 396/440
I realise this is incorrect, but I'm struggling to understand the process. Any help in giving me a eureka moment would be gratefully received. 
 A: Let's consider this:
$8$ pumps take $440L$ per $10 \min$. That means that $8$ pumps take $44L$ per $1\min$, which means that each pump takes $\frac{44}{8}L/\min$.
You want to know in how much minutes $x$, we have that $6$ pumps take $396L$. You have to multiply $6$ by $\frac{44}{8}$ to get the liters taken per minute by the $6$ pumps.
Now, you have 
$$\underbrace{\left(6\cdot\frac{44}{8}\right)}_{\text{liters taken per}\,\min}\cdot \underbrace{x}_{\text{number of}\,\min }=396$$
So that it leads to
$$x=\frac{396\cdot 8}{44\cdot 6}=12$$
And it takes $12\min$ for $6$ pumps to get $396L$.
A: When you have fewer pumps, you should not expect the time used for the entire task to be less, which is what you would get by multiplying by $6/8$ (which is less than $1$).
Instead, the number of pumps is proportional to the total rate of pumping, which is inversely proportional to the time any given task takes.
So when you change the rate, the time should be divided by the ration by which the rate changes, and your computation should be
$$ 10 \div \frac{6}{8} \times \frac{396}{440} $$
which is the same as
$$ 10 \times \frac{8}{6} \times \frac{396}{440} $$
A: If $8$ pumps process $440$ liters of water in $10$ minutes, then how much water does one pump process per minute?
A: perhaps you can give this a thought We know:
8 pumps do 10 minutes gives 440 L,
1 pump does 10 minutes gives 55 L (one eight of 440),
6 pumps do 10 minutes give 330 L (6 times 1 pump),
But we don't need 330 L, we need 396 L, so that's a factor 396/330 on the time they pump.
Therefore, 6 pumps do 10*396/330 minutes give (396/330)*330 (which is 330 L) Now simplify 10*396/330 minutes
A: It's easiest to understand this as a multi-part ratio: 
You get 440L using 8 pumps for 10 minutes--that translates to 
440/(8*10) Liters/pump-minute  = 11/2 Liters/pump-minute = 5.5 L/(p-m).  This is fixed.
Your question is then easily solved...set the fixed value to the # Liters/(number of pumps*number of minutes)  11/2 = 396/(6*minutes).
t = 12 minutes, as above.
