Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank.A. 1/3B. 1/2C. 1/4 D. 1E. 5/6

[tex]\begin{cases}\dfrac1a+\dfrac1b=\dfrac1{\frac65}=\dfrac56\\\\\dfrac1a+\dfrac1c=\dfrac1{\frac32}=\dfrac23\\\\\dfrac1b+\dfrac1c=\dfrac12\end{cases}[/tex]

Now,

[tex]\left(\dfrac1a+\dfrac1b\right)-\left(\dfrac1a+\dfrac1c\right)=\dfrac56-\dfrac23\implies\dfrac1b-\dfrac1c=\dfrac16[/tex]

Then

[tex]\left(\dfrac1b+\dfrac1c\right)+\left(\dfrac1b-\dfrac1c\right)=\dfrac12+\dfrac16\implies\dfrac2b=\dfrac23\implies b=3[/tex]

This means

[tex]\dfrac1a+\dfrac13=\dfrac56\implies\dfrac1a=\dfrac12\implies a=2[/tex]

and

[tex]\dfrac13+\dfrac1c=\dfrac12\implies\dfrac1c=\dfrac16\implies c=6[/tex]

Working together, all 3 pumps would operate at a rate of

[tex]\dfrac1a+\dfrac1b+\dfrac1c=1[/tex]

or 1 tank in 1 hour, so the answer is D.