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Updated on: 18 Jan 2024, 13:23
Originally posted by chicagocubsrule on 10 Nov 2009, 15:14.
Last edited by Bunuel on 18 Jan 2024, 13:23, edited 3 times in total.
Last edited by Bunuel on 18 Jan 2024, 13:23, edited 3 times in total.
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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. 1/3
B. 1/2
C. 2/3
D. 5/6
E. 1
Screenshot 2024-01-16 203959.png [ 38.99 KiB | Viewed 12719 times ]
A. 1/3
B. 1/2
C. 2/3
D. 5/6
E. 1
Attachment:
Screenshot 2024-01-16 203959.png [ 38.99 KiB | Viewed 12719 times ]
06 Mar 2010, 09:03
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You are adding the times, when you should be adding the rates.
Rates of:
A+B = 5/6 (tank per hour)
A+C = 2/3 (tank per hour)
B+C = 1/2 (tank per hour)
Combining these rates is like seeing "how much work would all of these pairs of machines do in 1 hour?"
(A+B)+(A+C)+(B+C) = 5/6 + 2/3 + 1/2 = 5/6 + 4/6 + 3/6 = 12/6 (tank per hour)
Therefore, 2A's, 2B's and 2C's working together would fill 2 tanks in an hour.
A single A, B, and C working together would fill 1 tank in 1 hour.
Rates of:
A+B = 5/6 (tank per hour)
A+C = 2/3 (tank per hour)
B+C = 1/2 (tank per hour)
Combining these rates is like seeing "how much work would all of these pairs of machines do in 1 hour?"
(A+B)+(A+C)+(B+C) = 5/6 + 2/3 + 1/2 = 5/6 + 4/6 + 3/6 = 12/6 (tank per hour)
Therefore, 2A's, 2B's and 2C's working together would fill 2 tanks in an hour.
A single A, B, and C working together would fill 1 tank in 1 hour.
11 Nov 2009, 05:52
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chicagocubsrule
can you explain a little bit about adding the equations?
Thanks
Thanks
Generally, if we are told that:
A hours is needed for worker A (pump A etc.) to complete the job --> the rate of A=\(\frac{1}{A}\);
B hours is needed for worker B (pump B etc.) to complete the job --> the rate of B=\(\frac{1}{B}\);
C hours is needed for worker C (pump C etc.) to complete the job --> the rate of C=\(\frac{1}{C}\);
You can see that TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).
Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.
Time*Rate=Distance
Time*Rate=Job
Also note that we can easily sum the rates:
If we are told that A is completing one job in 2 hours and B in 3 hours, thus A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.
Time needed for A and B working simultaneously to complete the job=\(\frac{A*B}{A+B}\) hours, which is reciprocal of the sum of their respective rates. (General formula for calculating the time needed for two workers working simultaneously to complete one job).
Time needed for A and C working simultaneously to complete the job=\(\frac{A*C}{A+C}\) hours.
Time needed for B and C working simultaneously to complete the job=\(\frac{B*C}{B+C}\) hours.
General formula for calculating the time needed for THREE workers working simultaneously to complete one job is: \(\frac{A*B*C}{AB+AC+BC}\) hours. Which is reciprocal of the sum of their respective rates: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}\).
We have three equations and three unknowns:
1. \(\frac{1}{A}+\frac{1}{B}=\frac{5}{6}\)
2. \(\frac{1}{A}+\frac{1}{C}=\frac{2}{3}\)
3. \(\frac{1}{B}+\frac{1}{C}=\frac{1}{2}\)
Now the long way is just to calculate individually three unknowns A, B and C from three equations we have. But as we just need the reciprocal of the sum of relative rates of A, B and C, knowing the sum of \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{AB+AC+BC}{ABC}\) would be fine, we just take the reciprocal of it and bingo, it would be just the value we wanted.
If we sum the three equations we'll get:
\(2*\frac{1}{A}+2*\frac{1}{B}+2*\frac{1}{C}=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2\)
\(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1\), now we just need to take reciprocal of 1, which is 1.
So the time needed for A, B, and C working simultaneously to complete 1 job is 1 hour.
Hope it helps.
