Problem Solving

BTGmoderatorDC wrote:John takes 15 hours to complete a certain job, while Bill takes only 6 hours to complete the same job. If Steve is faster than John but slower than Bill at completing the same job, then which of the following could be the time it takes the three men together, working at their constant, individual rates, to complete the job?

A. 2 hours, 30 minutes
B. 2 hours, 45 minutes
C. 3 hours, 20 minutes
D. 3 hours, 45 minutes
E. 4 hours, 10 minutes

Let's first see what happens if Steven works as fast as possible.
Since Bill can complete the job in 6 hours, Steven must complete the job in a little more than 6 hours.
For example, we COULD see what happens if Steven takes 6.000000000000001 hours to complete the job.
Unfortunately, 6.000000000000001 is an awful number to work with.
So, for convenience sake, let's just see what happens if it takes Steven 6 hours to complete the job

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

From Rule #1, we can conclude that:
- In ONE hour, John completes 1/15 of the job
- In ONE hour, Bill completes 1/6 of the job
- In ONE hour, Steven completes 1/6 of the job
So, COMBINED, the amount of the job completed after ONE hour = 1/15 + 1/6 + 1/6
= 2/30 + 5/30 + 5/30
= 12/30
= 2/5

Rule #2 tells us that, the total time to complete the job = 5/2 hours
In other words, when Steven works as FAST as possible, it takes the 3 men 5/2 hours to complete the job.
Of course, this calculation is based on Steven working at the SAME SPEED as Bill.
Since Steven is supposed to work SLOWER than Bill, the time it takes the 3 men to complete the job must be GREATER than 5/2 hours (aka 2 hours and 30 minutes)

IMPORTANT: We know that the correct answer must be GREATER than 2 hours and 30 minutes. So, we can eliminate answer choice A because it is too small.
We also know that, IF we were to also use Steven's SLOWEST work speed to calculate the upper limit for the time it takes all 3 men to complete the job, we'd be able to eliminate 3 more answer choices because they're too big.
Based on this, we should see that we can automatically eliminate the 3 biggest remaining answer choices (C, D, E), which leaves us with B

Answer: B

Cheers,
Brent

BTGmoderatorDC wrote:John takes 15 hours to complete a certain job, while Bill takes only 6 hours to complete the same job. If Steve is faster than John but slower than Bill at completing the same job, then which of the following could be the time it takes the three men together, working at their constant, individual rates, to complete the job?

A. 2 hours, 30 minutes
B. 2 hours, 45 minutes
C. 3 hours, 20 minutes
D. 3 hours, 45 minutes
E. 4 hours, 10 minutes

OA B

Source: Veritas Prep

John's rate is 1/15, and Bill's rate is 1/6.

If Steve is as fast as John, his rate is 1/15, and the combined rate would be 1/15 + 1/6 + 1/15 = 2/30 + 5/30 + 2/30 = 9/30 = 3/10. Therefore, it would take the three men 1/(3/10) = 10/3 = 3 â…“ = 3 hours and 20 minutes to complete the job.

On the other hand, if Steve is as fast as Bill, his rate is 1/6, and the combined rate would be 1/15 + 1/6 + 1/6 = 2/30 + 5/30 + 5/30 = 12/30 = 2/5. Therefore, it would take the three men 1/(2/5) = 5/2 = 2 1/2 = 2 hours and 30 minutes to complete the job.

Since Steve is faster than John but slower than Bill, it will take them between 2 hours and 30 minutes and 3 hours and 20 minutes to complete the job. We see that of all the answer choices, only choice B, 2 hours and 45 minutes, satisfies the criteria.

Answer: B