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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?

Answer Choices

A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{2}{3}$
D. $\frac{5}{6}$
E. $1$

Steps

Explanation	Calculations	Help
$We name the variables for the pumps’ rates.$	$a = rate of pump A in tanks per hour b = rate of pump B in tanks per hour c = rate of pump C in tanks per hour$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "Pumps A and B, operating simultaneously, can fill a certain tank in \frac{6}{5} hours" into a work equation.$	$(a + b) \times \frac{6}{5} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We translate "Pumps A and C, operating simultaneously, can fill the tank in \frac{3}{2} hours" into a work equation.$	$(a + c) \times \frac{3}{2} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We translate "Pumps B and C, operating simultaneously, can fill the tank in 2 hours" into a work equation.$	$(b + c) \times 2 = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We solve each equation for the sum of two rates.$	$a + b = \frac{1}{\frac{6}{5}} = \frac{5}{6} a + c = \frac{1}{\frac{3}{2}} = \frac{2}{3} b + c = \frac{1}{2}$	Theory & Tactics Method Card EQUA1-A Click to view full details
$We add the three simplified equations to find twice the sum of all three rates.$	$2 (a + b + c) = \frac{5}{6} + \frac{2}{3} + \frac{1}{2}$	Theory & Tactics Method Card EQUA3-B Click to view full details
$We simplify the right side by converting to a common denominator.$	$\frac{5}{6} + \frac{2}{3} + \frac{1}{2} = \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{12}{6} = 2$
$We divide both sides by 2 to get the combined rate.$	$a + b + c = \frac{2}{2} = 1$	Theory & Tactics Method Card EQUA1-C Click to view full details
$The time to fill one tank is the reciprocal of the combined rate.$	$Time = \frac{1}{a + b + c} = \frac{1}{1} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details

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Final Answer

1

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?

Answer Choices

A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{2}{3}$
D. $\frac{5}{6}$
E. $1$

Steps

Explanation	Calculations	Help
$We name the variables for the pumps’ rates.$	$a = rate of pump A in tanks per hour b = rate of pump B in tanks per hour c = rate of pump C in tanks per hour$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "Pumps A and B, operating simultaneously, can fill a certain tank in \frac{6}{5} hours" into a work equation.$	$(a + b) \times \frac{6}{5} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We translate "Pumps A and C, operating simultaneously, can fill the tank in \frac{3}{2} hours" into a work equation.$	$(a + c) \times \frac{3}{2} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We translate "Pumps B and C, operating simultaneously, can fill the tank in 2 hours" into a work equation.$	$(b + c) \times 2 = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details
$We solve each equation for the sum of two rates.$	$a + b = \frac{1}{\frac{6}{5}} = \frac{5}{6} a + c = \frac{1}{\frac{3}{2}} = \frac{2}{3} b + c = \frac{1}{2}$	Theory & Tactics Method Card EQUA1-A Click to view full details
$We add the three simplified equations to find twice the sum of all three rates.$	$2 (a + b + c) = \frac{5}{6} + \frac{2}{3} + \frac{1}{2}$	Theory & Tactics Method Card EQUA3-B Click to view full details
$We simplify the right side by converting to a common denominator.$	$\frac{5}{6} + \frac{2}{3} + \frac{1}{2} = \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{12}{6} = 2$
$We divide both sides by 2 to get the combined rate.$	$a + b + c = \frac{2}{2} = 1$	Theory & Tactics Method Card EQUA1-C Click to view full details
$The time to fill one tank is the reciprocal of the combined rate.$	$Time = \frac{1}{a + b + c} = \frac{1}{1} = 1$	Theory & Tactics Method Card TRANS3-D Click to view full details

Scroll horizontally to view all columns

Final Answer

1