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 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?

GMAT Panda · Accepted Answer

1 We name the variables for the pumps’ rates. a = rate of pump A in tanks per hour b = rate of pump B in tanks per hour c = rate of pump C in tanks per hour We translate **"Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours"** into a work equation. (a + b) 6/5 = 1 We translate **"Pumps A and C, operating simultaneously, can fill the tank in 3/2 hours"** into a work equation. (a + c) 3/2 = 1 We translate "Pumps B and C, operating simultaneously, can fill the tank in 2 hours" into a work equation. (b + c) 2 = 1 We solve each equation for the sum of two rates. a + b = 16/5 = 5/6 a + c = 13/2 = 2/3 b + c = 1/2 We add the three simplified equations to find twice the sum of all three rates. 2(a + b + c) = 5/6 + 2/3 + 1/2 We simplify the right side by converting to a common denominator. 5/6 + 2/3 + 1/2 = 5/6 + 4/6 + 3/6 = 12/6 = 2 We divide both sides by 2 to get the combined rate. a + b + c = 2/2 = 1 The time to fill one tank is the reciprocal of the combined rate. Time = 1/a + b + c = 1/1 = 1

Explanation	Calculations	Help
$We name the variables for the pumps’ rates.$	$a = rate of pump A in tanks per hour b = rate of pump B in tanks per hour c = rate of pump C in tanks per hour$
$We translate "Pumps A and B, operating simultaneously, can fill a certain tank in \frac{6}{5} hours" into a work equation.$	$(a + b) \times \frac{6}{5} = 1$
$We translate "Pumps A and C, operating simultaneously, can fill the tank in \frac{3}{2} hours" into a work equation.$	$(a + c) \times \frac{3}{2} = 1$
$We translate "Pumps B and C, operating simultaneously, can fill the tank in 2 hours" into a work equation.$	$(b + c) \times 2 = 1$
$We solve each equation for the sum of two rates.$	$a + b = \frac{1}{\frac{6}{5}} = \frac{5}{6} a + c = \frac{1}{\frac{3}{2}} = \frac{2}{3} b + c = \frac{1}{2}$
$We add the three simplified equations to find twice the sum of all three rates.$	$2 (a + b + c) = \frac{5}{6} + \frac{2}{3} + \frac{1}{2}$
$We simplify the right side by converting to a common denominator.$	$\frac{5}{6} + \frac{2}{3} + \frac{1}{2} = \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{12}{6} = 2$
$We divide both sides by 2 to get the combined rate.$	$a + b + c = \frac{2}{2} = 1$
$The time to fill one tank is the reciprocal of the combined rate.$	$Time = \frac{1}{a + b + c} = \frac{1}{1} = 1$

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