This is the Problem of the Week answer for this post.

# Part #1

Both Don and Juan and are given a task on which they must work together to complete in a timely fashion. It takes Don 75 minutes to do the job by himself. It takes Juan 60 minutes to do the job by himself.

1. First we must figure out what portion of the task Don will do compared to what portion Juan will do.  Their portions will depend on the speed at which they work compared to each other.
Name Time Comparative Speed
Don 75 75/75
Juan 60 75/60
2. Now to figure out the portion that Don and Juan will be allotted to work together simultaneously to get the job done.  To do this, you must divide their Comparitive Speed by the sum of both their Comparitive Speeds.
Name Time Comparative Speed Portion
Don 75 75/75 = 1 1/1 + 1.25 = 4/9
Juan 60 75/60 = 1.25 1.25/1 + 1.25 = 5/9
3. Now it is time to multiply their times by their portions to get the amount of time it will take the task to get done.
Name Time Comparative Speed Portion New Time
Don 75 75/75 = 1 1/1 + 1.25 = 4/9 75 × 4/9 = 100/3
Juan 60 75/60 = 1.25 1.25/1 + 1.25 = 5/9 60 × 5/9 = 100/3

If done correctly, the New Time for each person should be the same as shown above. Therefore, the answer in this case is 33 minutes and 20 seconds.

4. To come up with a formula for New Time, let’s step through the process of getting the new time for Don’s row, substituting 75 for A and 60 for B
1. Time:  A = 75
2. Comparative Speed:  A / A
3. Portion:  (A / A) / (A / A + A / B)
4. New Time:  A × (A / A) / (A / A + A / B) = A / (A / A + A / B) = 1 / (1 / A + 1 / B) = AB / (B + A)

As you can see in step three, the amount of time that it would take for both Don and Juan to accomplish the task while working together is 33 minutes and 20 seconds.  A formula that can be used to figure this out for any two times is as follows:
AB / (A + B)

# Part #2

Don, Juan, and Guadalupe are given a task on which they must work together to complete in a timely fashion. It takes Don 75 minutes to do the job by himself. It takes Juan 60 minutes to do the job by himself.  It takes Guadalupe 68 minutes to do the same task by herself.

1. First we must figure out what portion each person will do according to how fast the do the task compared to everyone else. Therefore, let’s figure out some number to compare each person’s speed:
Name Time Comparative Speed
Don 75 68/75
Juan 60 68/60

NOTE: The time that appears in the denominator of the Comparative can belong to any of the three people.

2. Now to figure out the portion that Don, Juan, and Guadalupe will be allotted to work together simultaneously to get the job finished at the same time.  To do this, you must divide their Comparitive Speed by the sum of both their Comparitive Speeds.
Name Time Comparative Speed Portion
Don 75 68/75 (68 / 75) / (68 / 75 + 68 / 60 + 68 / 68) = 4080/13680
Juan 60 68/60 (68 / 60)(68 / 75 + 68 / 60 + 68 / 68) = 5100/13680
Guadalupe 68 68/68 (68 / 68)(68 / 75 + 68 / 60 + 68 / 68) = 4500/13680
3. Now it is time to multiply their times by their portions to get the amount of time it will take the task to get done.
Name Time Comparative Speed Portion New Time
Don 75 68/75 4080/13680 75 × 4080/13680 ≈ 22.368
Juan 60 68/60 5100/13680 60 × 5100/13680 ≈ 22.368
Guadalupe 68 68/68 4500/13680 68 × 4500/13680 ≈ 22.368

If done correctly, the New Time for each person should be the same as shown above. Therefore, the answer in this case is about 22 minutes and 22 seconds.

4. To come up with a formula for New Time, let’s step through the process of getting the new time for Guadalupe’s row, substituting 75 for A, 60 for B, 68 for C
1. Time:  C = 75
2. Comparative Speed:  C / C
3. Portion:
1. (C / C) / (C / A + C / B + C / C)
2. (1 / C) / (1 / A + 1 / B + 1 / C)
3. (1 / C) / ((BC + AC + AB) / ABC)
4. ABC / (BCC + ACC + ABC)
5. AB / (BC + AC + AB)
4. New Time:  C × AB / (BC + AC + AB)

As you can see in step three, the amount of time that it would take for both Don, Juan, and Guadalupe to accomplish the task while working together is about 22 minutes and 22 seconds.  A formula that can be used to figure this out for any three times is as follows:
ABC / (BC + AC + AB)

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#### POW – Working Together | Chris West's Blog · February 9, 2012 at 10:34 AM

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