Problem Solving

Please see attached image below: I understand the following (and feel free to correct me if i'm wrong)

Pump A empties Pool in A minutes
Pump B empties Pool in B minutes

so in 1 minute: A empties 1/A of pool
and in 1 minutes: B empties 1/B of pool

so if they started the same time, it would take:
1/A + 1/B = (A+B)/AB -> AB/(A+B) minutes to empty the pool.

BUT, since pump A started 1 minute earlier I had to find the time it would take for both pumps together would pump the same pool AB/(A+B) and multiply by how much of the pool was left after 1 minute of Pump A.

Here's where I get confused.

the explanation states that after 1 minute of Pump A, which is 1/A pumped out, there's (A-1)/A left.
I DON'T GET THIS PART! I feel like i'm missing something very basic.

Do I have to assume that since pumps A & B are pumping the same pool, we can say that the "pool" is "1"?

Seems like (A-1)/A = A/A - 1/A = 1 - 1/A
oy... i hope im not confusing anyone.

forgot image.

additionally, if someone has a much smarter, efficient way to solve this... i would take that over any day. but i'd still love to hear an answer to my specific question.

thank you!

Think of it in numbers. If A takes 10 minutes, after one minute there are nine minutes left. That's 9/10 of the original time or (10-1)/10 which is the same as (A-1)/A

In your scenario, that would be 1- 1/10, which is also equal to 9/10.

thank you.. that was easy enough.

Pump A can empty a pool in A minutes, and pump B can empty the same pool in B minutes. Pump A begins emptying the pool for 1 minute before pump B joins. Beginning from the time pump A starts, how many minutes will it take to empty the pool?

Let pool = 6 units.
Let time for A = 2 minutes and time for B = 3 minutes.
Rate for A = w/t = 6/2 = 3 units per minute.
Rate for B = w/t = 6/3 = 2 units per minute.
Combined rate for A+B = 3+2 = 5 units per minute.
In 1 minute, A pumps out 3 units.
Remainder of pool = 6-3 = 3 units.
Time for A+B to finish = w/r = 3/5 minutes.
Total time = 1 + 3/5 = 8/5. This is our target.

Now we plug A=2 and B=3 into the answers to see which yields our target of 8/5.

Only answer choice B works:
A(B + 1)/ (A + B) = 2(3+1)/ (2+3) = 8/5.

The correct answer is B.