When three pumps, A, B, and C, are running together, they can pump 3400 gal per hour. When only A, and B are running, 1900 gal per hour can be pumped. When only A and C are running, 2300 gal per hour can be pumped. What is the pumping capacity of each pump?
What is the pumping capacity of A___GAL PER HOUR?
What is the pumping capacity of B___GAL PER HOUR?
What is the pumping capacity of C___GAL PER HOUR?
1/A + 1/B + 1/C = 3400 , #1
1/A + 1/B = 1900
(A+B)/(AB) = 1900
1900AB = A+B
1900AB - B = A
B(1900A - 1) = A
B = A/(1900A - 1) , #2
1/A + 1/C = 2300
A+C = 2300AC
...
C = A/(2300A - 1) , #3
SUB #2 and #3 into #1
1/A + 1/(A/(1900A - 1)) + 1/A/(2300A - 1)) = 3400
1/A + (1900A - 1)/A + (2300A - 1)/A = 3400
1 + 1900A - 1 + 2300A - 1 = 3400A
800A = 1
A = 1/800
Subbing back into #2 you will get
B = 1/1100
and into #3 you will get
C = 1/1500
A's rate is 800 gal/hr
B's rate is 1100 gal/hr
C's rate is 1500 gal/hr
Thank you
To determine the pumping capacity of each pump, we can set up a system of equations based on the given information.
Let's denote the pumping capacity of pump A as A_GPH (gallons per hour), the pumping capacity of pump B as B_GPH, and the pumping capacity of pump C as C_GPH.
From the information provided, we have the following equations:
Equation 1: A_GPH + B_GPH + C_GPH = 3400 (when all three pumps are running together)
Equation 2: A_GPH + B_GPH = 1900 (when only pumps A and B are running together)
Equation 3: A_GPH + C_GPH = 2300 (when only pumps A and C are running together)
To solve these equations, we can use a method called elimination or substitution.
Let's start with the elimination method:
Taking Equation 2 (A_GPH + B_GPH = 1900) and Equation 3 (A_GPH + C_GPH = 2300), we can subtract Equation 3 from Equation 2 to eliminate the A_GPH term:
(A_GPH + B_GPH) - (A_GPH + C_GPH) = 1900 - 2300
B_GPH - C_GPH = -400
Now, let's substitute this value into Equation 1 (A_GPH + B_GPH + C_GPH = 3400) to solve for B_GPH:
A_GPH + (-400) + C_GPH = 3400
A_GPH + C_GPH - 400 = 3400
A_GPH + C_GPH = 3400 + 400
A_GPH + C_GPH = 3800
We can now substitute the value of A_GPH + C_GPH from this equation into Equation 3 (A_GPH + C_GPH = 2300):
3800 = 2300
2A_GPH = 3800 - 2300
2A_GPH = 1500
A_GPH = 1500 / 2
A_GPH = 750
Now that we have the value of A_GPH, we can substitute it back into Equation 2 to solve for B_GPH:
750 + B_GPH = 1900
B_GPH = 1900 - 750
B_GPH = 1150
Finally, we can substitute the values of A_GPH and B_GPH into Equation 1 to solve for C_GPH:
750 + 1150 + C_GPH = 3400
C_GPH = 3400 - 750 - 1150
C_GPH = 1500
Therefore, the pumping capacity of pump A is 750 GAL PER HOUR, the pumping capacity of pump B is 1150 GAL PER HOUR, and the pumping capacity of pump C is 1500 GAL PER HOUR.