Pumps A, B, and C operate at their respective constant rates. Pumps A

Generally, if we are told that:
A hours is needed for worker A (pump A etc.) to complete the job --> the rate of A=\(\frac{1}{A}\);
B hours is needed for worker B (pump B etc.) to complete the job --> the rate of B=\(\frac{1}{B}\);
C hours is needed for worker C (pump C etc.) to complete the job --> the rate of C=\(\frac{1}{C}\);

You can see that TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).

Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.

Time*Rate=Distance
Time*Rate=Job

Also note that we can easily sum the rates:
If we are told that A is completing one job in 2 hours and B in 3 hours, thus A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.

Time needed for A and B working simultaneously to complete the job=\(\frac{A*B}{A+B}\) hours, which is reciprocal of the sum of their respective rates. (General formula for calculating the time needed for two workers working simultaneously to complete one job).
Time needed for A and C working simultaneously to complete the job=\(\frac{A*C}{A+C}\) hours.

Time needed for B and C working simultaneously to complete the job=\(\frac{B*C}{B+C}\) hours.

General formula for calculating the time needed for THREE workers working simultaneously to complete one job is: \(\frac{A*B*C}{AB+AC+BC}\) hours. Which is reciprocal of the sum of their respective rates: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}\).

We have three equations and three unknowns:
1. \(\frac{1}{A}+\frac{1}{B}=\frac{5}{6}\)

2. \(\frac{1}{A}+\frac{1}{C}=\frac{2}{3}\)

3. \(\frac{1}{B}+\frac{1}{C}=\frac{1}{2}\)

Now the long way is just to calculate individually three unknowns A, B and C from three equations we have. But as we just need the reciprocal of the sum of relative rates of A, B and C, knowing the sum of \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{AB+AC+BC}{ABC}\) would be fine, we just take the reciprocal of it and bingo, it would be just the value we wanted.

If we sum the three equations we'll get:
\(2*\frac{1}{A}+2*\frac{1}{B}+2*\frac{1}{C}=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2\)

\(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1\), now we just need to take reciprocal of 1, which is 1.

So the time needed for A, B, and C working simultaneously to complete 1 job is 1 hour.

Hope it helps.