Combined Rates

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

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