GMAT Practice Question: Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating...
Question
Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in \frac{6}{5} hours; Pumps A and C, operating simultaneously, can fill the tank in \frac{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?
- \frac{1}{3}
- \frac{1}{2}
- \frac{2}{3}
- \frac{5}{6}
- 1
Topics: word problems, rates
Solution
Step 1
We name the variables for the pumps’ rates.
a = \text{rate of pump A in tanks per hour} b = \text{rate of pump B in tanks per hour} c = \text{rate of pump C in tanks per hour}
Step 2
We translate **"Pumps A and B, operating simultaneously, can fill a certain tank in \frac{6}{5} hours"** into a work equation.
(a + b) \times \frac{6}{5} = 1
Step 3
We translate **"Pumps A and C, operating simultaneously, can fill the tank in \frac{3}{2} hours"** into a work equation.
(a + c) \times \frac{3}{2} = 1
Step 4
We translate \textbf{"Pumps B and C, operating simultaneously, can fill the tank in 2 hours"} into a work equation.
(b + c) \times 2 = 1
Step 5
We solve each equation for the sum of two rates.
a + b = \frac{1}{\frac{6}{5}} = \frac{5}{6} a + c = \frac{1}{\frac{3}{2}} = \frac{2}{3} b + c = \frac{1}{2}
Step 6
We add the three simplified equations to find twice the sum of all three rates.
2(a + b + c) = \frac{5}{6} + \frac{2}{3} + \frac{1}{2}
Step 7
We simplify the right side by converting to a common denominator.
\frac{5}{6} + \frac{2}{3} + \frac{1}{2} = \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{12}{6} = 2
Step 8
We divide both sides by 2 to get the combined rate.
a + b + c = \frac{2}{2} = 1
Step 9
The time to fill one tank is the reciprocal of the combined rate.
\text{Time} = \frac{1}{a + b + c} = \frac{1}{1} = 1
Answer
1
