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12 Nov 2020, 09:30
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Irising
Yes, we can sum rates but not times.
THEORY
There are several important things you should know to solve work problems:
1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.
\(time*speed=distance\) <--> \(time*rate=job \ done\). For example when we are told that a man can do a certain job in 3 hours we can write: \(3*rate=1\) --> \(rate=\frac{1}{3}\) job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then \(5*(2*rate)=1\) --> so rate of 1 printer is \(rate=\frac{1}{10}\) job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then \(3*(2*rate)=12\) --> so rate of 1 printer is \(rate=2\) pages per hour;
So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).
2. We can sum the rates.
If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is \(rate_a=\frac{job}{time}=\frac{1}{2}\) job/hour and B's rate is \(rate_b=\frac{job}{time}=\frac{1}{3}\) job/hour. Combined rate of A and B working simultaneously would be \(rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\) job/hour, which means that they will complete \(\frac{5}{6}\) job in one hour working together.
3. For multiple entities: \(\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}\), where \(T\) is time needed for these entities to complete a given job working simultaneously.
For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;
Then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}\), where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.
For two and three entities (workers, pumps, ...):
General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:
Given that \(t_1\) and \(t_2\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}\)).
General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:
\(T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3}\) hours.
17. Work/Rate Problems
- Theory
- Questions
Work Rate Problems - All in One Topic!
Work Word Problems Made Easy
How to Solve Relative Rate of Work Questions on the GMAT
DS Questions
PS Questions
On other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread
avigutman
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Joined: 17 Jul 2019
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GMAT 1: 780 Q51 V45
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Expert reply
20 Mar 2021, 20:47
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Video solution from Quant Reasoning starts at 7:30
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
10 Feb 2022, 18:41
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Bunuel
slow pump works at a rate of 1x, fast pump works at a rate of 1.5x
slow pump does 1/3 of the job in 4 hours, fast pump does 2/3 of the job in 4 hours
Thus, for fast pump to accomplish the remaining 1/3 of the job, it would take 6 hours.
Where does this go wrong?
