Combined Rates

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kamalakarthi: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Mon Jan 31, 2011 10:08 pmFollowed by: 2 members

by kamalakarthi » Sat May 13, 2017 5:18 am

Hi, In the attached question, My approach was to calculate A+B+C and subtract A+C to come to B's time. Can you please help me on What am I missing in concept.

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Source: Beat The GMAT — Problem Solving | Sat May 13, 2017 5:18 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat May 13, 2017 5:23 am

kamalakarthi wrote:Hi, In the attached question, My approach was to calculate A+B+C and subtract A+C to come to B's time. Can you please help me on What am I missing in concept.

I posted an approach for a very similar problem here:
https://www.beatthegmat.com/work-rates-c ... 71522.html

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kamalakarthi: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Mon Jan 31, 2011 10:08 pmFollowed by: 2 members

by kamalakarthi » Sat May 13, 2017 7:06 am

Thank you Mitch. Is it fair to say that I will be able to solve all the rate related questions with the common unit of work. In this case it is 18. My concern is what if I am in situation where I am not able to come to a common unit of work.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat May 13, 2017 9:05 am

A and B together can complete a task in 10 hours. B and C together can complete the same task in 18 hours. C and A together can complete the same task in 15 hours. How many hours will it take B, working alone, to complete the task?

A) 15
B) 18
C) 20
D) 22.5
E) 25

For work questions, there are two useful rules:

Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour

Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.

Let's use these rules to solve the question. . . .

Let A = amount of work that A can do in ONE HOUR
Let B = amount of work that B can do in ONE HOUR
Let C = amount of work that C can do in ONE HOUR

A and B together can complete a task in 10 hours.
Applying rule #1, we can conclude that, in ONE HOUR, they can complete 1/10 of the task.
In other words, A + B = 1/10

B and C together can complete the same task in 18 hours
Applying rule #1, we can conclude that, in ONE HOUR, they can complete 1/18 of the task.
In other words, B + C = 1/18

C and A together can complete the same task in 15 hours.
So, A + C = 1/15

We have:
A + B = 1/10
B + C = 1/18
A + C = 1/15

Add ALL THREE equations together to get:
2A + 2B + 2C = 1/10 + 1/18 + 1/15
Find common denominators: 2A + 2B + 2C = 18/180 + 10/180 + 12/180
Simplify: 2(A + B + C) = 40/180
Divide both sides by 2 to get: A + B + C = 20/180
Our job is to find the value of B

We already know that A + C = 1/15
So, take A + B + C = 20/180 and replace A + C with 1/15
We get: 1/15 + B = 20/180
Rewrite as: 12/180 + B = 20/180
Solve: B = 8/180
In other words, B can complete 8/180 of the job in ONE HOUR

Now apply rule #2 to see that the time to complete the ENTIRE TASK = 180/8 hours.
180/8 = 22.5 hours

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 15, 2017 3:01 am

kamalakarthi wrote:Thank you Mitch. Is it fair to say that I will be able to solve all the rate related questions with the common unit of work. In this case it is 18. My concern is what if I am in situation where I am not able to come to a common unit of work.

Generally, if the job is undefined, we can plug in ANY VALUE for the job.
To make the math easier, plug in a value divisible by the given times.

A and B together can complete a task in 10 hours. B and C together can complete the same task in 18 hours. C and A together can complete the same task in 15 hours. How many hours will it take B, working alone, to complete the task?

A) 15
B) 18
C) 20
D) 22.5
E) 25

Let the job = the LCM of 10, 18 and 15 = 90.

Since A and B take 10 hours to complete the task, the rate for A+B = w/t = 90/10 = 9 units per hour.
Since B and C take 18 hours to complete the task, the rate for B+C = w/t = 90/18 = 5 units per hour.
Since A and C take 15 hours to complete the task, the rate for A+C = w/t = 90/15 = 6 units per hour.

Thus:
(A+B) + (B+C) + (A+C) = 9+5+6
2A + 2B + 2C = 20
A+B+C = 10 units per hour.

Subtracting A+C = 6 from A+B+C = 10, we get:
(A+B+C) - (A+C) = 10-6
B = 4 units per hour.

Thus:
At a rate of 4 units per hour, the time for B to complete the 90-unit task = w/r = 90/4 = 45/2 = 22.5 hours.

The correct answer is D.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri May 19, 2017 5:01 am

A and B together can complete a task in 10 hours. B and C together can complete the same task in 18 hours. C and A together can complete the same task in 15 hours. How many hours will it take B, working alone, to complete the task?

A) 15
B) 18
C) 20
D) 22.5
E) 25

We can let a = the time it takes A to complete the job, b = the time it takes B to complete the job, and c = the time it takes C to complete the job. Thus, the rate of A is 1/a, the rate of B is 1/b, and the rate of C is 1/c.

Since A and B can complete the task in 10 hours:

1/a + 1/b = 1/10

Since B and C can compete the task in 18 hours:

1/b + 1/c = 1/18

Since C and A can complete the task in 15 hours:

1/c + 1/a = 1/15

We can add the first 2 equations together and we have:

1/a + 2/b + 1/c = 1/10 + 1/18

1/a + 2/b + 1/c = 9/90 + 5/90

1/a + 2/b + 1/c = 14/90

To determine how long it takes B to complete the task alone, we want to determine the rate for B, which is 1/b. We can subtract 1/c + 1/a = 1/15 from 1/a + 2/b + 1/c = 14/90:

2/b = 14/90 - 1/15

2/b = 14/90 - 6/90

2/b = 8/90

1/b = 4/90

Since the rate of B is 4/90, B's time is 1/(4/90) = 90/4 = 22.5 hours.

Answer: D

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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