Combined rate formula? : GMAT Quantitative Section
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# Combined rate formula?

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Manager
Joined: 26 Feb 2013
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18 Jun 2013, 22:24
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I remember having seen a formula for rates

$$\frac{A*B}{A+B}$$ but I can't remember what it is for (if anything at all). Can someone clarify this for me?

Thanks
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19 Jun 2013, 00:09
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Skag55 wrote:
I remember having seen a formula for rates

$$\frac{A*B}{A+B}$$ but I can't remember what it is for (if anything at all). Can someone clarify this for me?

Thanks

hi,

If A can finish a work in X time and B can finish the
same work in Y time then both of them together can
finish that work in (X*Y)/ (X+Y) time.

If A can finish a work in X time and A & B together can
finish the same work in S time then B can finish that
work in (XS)/(X-S) time.

If A can finish a work in X time and B in Y time and C in
Z time then all of them working together will finish the
work in (XYZ)/ (XY +YZ +XZ) time

kudos if it helped
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19 Jun 2013, 01:13
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shaileshmishra wrote:
Skag55 wrote:
I remember having seen a formula for rates

$$\frac{A*B}{A+B}$$ but I can't remember what it is for (if anything at all). Can someone clarify this for me?

Thanks

hi,

If A can finish a work in X time and B can finish the
same work in Y time then both of them together can
finish that work in (X*Y)/ (X+Y) time.

If A can finish a work in X time and A & B together can
finish the same work in S time then B can finish that
work in (XS)/(X-S) time.

If A can finish a work in X time and B in Y time and C in
Z time then all of them working together will finish the
work in (XYZ)/ (XY +YZ +XZ) time

kudos if it helped

...kudos if it deserves it.

Though I think it does, cause it covers more than I asked and it's nicely laid out!
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19 Jun 2013, 01:16
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Skag55 wrote:
I remember having seen a formula for rates

$$\frac{A*B}{A+B}$$ but I can't remember what it is for (if anything at all). Can someone clarify this for me?

Thanks

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.

Some work problems with solutions:
time-n-work-problem-82718.html?hilit=reciprocal%20rate
facing-problem-with-this-question-91187.html?highlight=rate+reciprocal
what-am-i-doing-wrong-to-bunuel-91124.html?highlight=rate+reciprocal
gmat-prep-ps-93365.html?hilit=reciprocal%20rate
a-good-one-98479.html?hilit=rate
solution-required-100221.html?hilit=work%20rate%20done
work-problem-98599.html?hilit=work%20rate%20done

All DS work/rate problems to practice: search.php?search_id=tag&tag_id=46
All PS work/rate problems to practice: search.php?search_id=tag&tag_id=66

Hope it helps.
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19 Jun 2013, 01:19
Thanks Bunuel, helpful and thorough as always!
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11 Apr 2015, 17:15
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12 Apr 2015, 18:29
Hi All,

Here's an example of a standard "2 entity" question that can be solved with the Work Formula:

Pump A can fill a swimming pool in 2 hours. Pump B can fill the same pool in 6 hours. If the pool is empty and both Pumps are started at the same time, then how many hours does it take for the two Pumps to fill the pool?

Work Formula = (A)(B)/(A+B)

Since we have the rates of the two pumps, we can plug them directly into the Work Formula:

(2)(6)/(2+6) = 12/8 = 1.5 hours to fill the pool when working together.

Variations on this question type can include having you calculate one of the 'starting rates' or setting up a 'system' of Work equations to solve for both starting rates.

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28 Apr 2016, 22:37
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Combined rate formula?   [#permalink] 28 Apr 2016, 22:37
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