OG PS #126

[diegocuenca]

It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?

1/A + 1/B = 1/T

1/4 + 1/3 = 1/T => 3/12 + 4/12 = 7/12, from here I had troubling knowing what to do with this information. I ultimately chose 7/12, knowing it was probably wrong, because my mind went blank. The solution states that the first machine machine can complete 1/4 of the order in one hours and the second can complete 1/3 of the order in one hours. That clicked for me, because that was the information I was trying to come up with. However, it will take 12/7 = 1 5/7 hours for the two machines to complete the order together. I don't understand how they went from 7/12 working together to 12/7 hours to complete the work.

[blackjack]

Hi,
Like you said, the combined rate is 1/4 + 1/3 = 7/12 work per hour.

So time taken to do '1' work is work/rate= 1/(7/12)= 12/7

[tpr-becky]

In questions that talk about "an order" or "a job" sometimes it is helpful to name a number bigger than one to avoid the fractions. For instance in this problem "a large ... order" could be reworked to say the order had 24 pieces. Another key to rate problems is that each situation gets its own rate formula - in this one we have three (the first machine, the second machine and working together).

Thus if the first machine can do the 24 pieces in 4 hours then d=r x t and 24 = r(4) so the rate of the first machine is 6 pieces per hour.

The second machine can do the 24 pieces in 3 hours - 24 = r(3) so the rate of the second machine is 8.

When machines work together you add their rates. so 24 = (6+8)(t) to get 24/14 = time, which reduces to 12/7.

[Ashley@VeritasPrep]

It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?

1/A + 1/B = 1/T

1/4 + 1/3 = 1/T => 3/12 + 4/12 = 7/12, from here I had troubling knowing what to do with this information. I ultimately chose 7/12, knowing it was probably wrong, because my mind went blank. The solution states that the first machine machine can complete 1/4 of the order in one hours and the second can complete 1/3 of the order in one hours. That clicked for me, because that was the information I was trying to come up with. However, it will take 12/7 = 1 5/7 hours for the two machines to complete the order together. I don't understand how they went from 7/12 working together to 12/7 hours to complete the work.

Hey!

I think the reason for having trouble knowing what to do with the information above is pretty definitely that that formula is not very explicit, i.e. it's not immediately clear what the variables stand for nor is it intuitive why in the world that statement (1/A + 1/B = 1/T) would be true. You may be correctly sensing that I'm a little biased against that formula -- I freely admit it. I actually recommend against using it, for two reasons. 1, it's sufficiently not-directly-intuitive that students ALL THE TIME encounter the exact problem you encountered, where they get to the end and don't know what to do with the information they've come to, because they don't know what that information represents -- like the formula almost *invites* you to not understand the math. 2, the formula only even works at all in the most basic rate problems -- if the problem throws you any curveball (like one worker starting later than another or stopping earlier than another), then the formula will be useless and you'll HAVE to work through the math in a more intuitive way.

The approach I recommend:

Take everything down to per-hour productivity. Here, your first machine takes 4 hours for 1 order, so it can do 1/4 order per hour. Machine B takes 3 hours for 1 order, so it can do 1/3 order per hour.

Once I've got their per-hourly rates, I add. If they work together, 1/4 order + 1/3 order will get done, i.e. 7/12 order per hour.

Then you've essentially got a proportion:

7/12 order = 1 order
d 1 hour ddd x hours

Cross multiplying gives you that (7/12)x = 1. So x = 1/(7/12) = 12/7.

You can see how this approach basically takes me through the same steps as the formula does (i.e. getting to 1/4 and 1/3 and adding them, etc.), but I think this approach keeps track of where all those numbers are coming from and what they mean, so there's a lot less room for confusion.

PS - Becky's approach is also a good one!

[winniethepooh]

Hey.
There's another approach, using the same formula(but in a simplified form):
Using the same formula:
1/s + 1/t = 1/h(Where s and t are two works and h is the combined rate.
s * t/s + t =h
Therefore, h = s * t / s + t
For the sum,
h= 4 * 3 / 4 + 3 = 12/7.
Simple, isn't it?

[Scott@TargetTestPrep]

It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?

1/A + 1/B = 1/T

1/4 + 1/3 = 1/T => 3/12 + 4/12 = 7/12, from here I had troubling knowing what to do with this information. I ultimately chose 7/12, knowing it was probably wrong, because my mind went blank. The solution states that the first machine machine can complete 1/4 of the order in one hours and the second can complete 1/3 of the order in one hours. That clicked for me, because that was the information I was trying to come up with. However, it will take 12/7 = 1 5/7 hours for the two machines to complete the order together. I don't understand how they went from 7/12 working together to 12/7 hours to complete the work.

Solution:

We can classify this problem as a "combined worker" problem. To solve this type of problem we can use the formula:

Work (done by worker 1) + Work (done by worker 2) = Total Work Completed

It takes machine one 4 hours to complete a job, so the rate of machine one is ¼. It takes machine two 3 hours to complete a job, so the rate of machine two is 1/3. Since we know they are both working together to complete the job, we can label this unknown time as "t," which is the total amount of time to complete the job. Since rate x time = work, we can multiply to get the work completed for each machine.

We can put all the above information into a rate-time-work table.



Finally, we can plug our two work values into the combined work formula and determine t. Since one job is completed, the total work completed is 1.

Work (done by machine 1) + Work (done by machine 2) = Total Work Completed

(1/4)t + (1/3)t = 1

Multiplying the entire equation by 12 gives us:

3t + 4t = 12

7t = 12

t = 12/7 = 1 5/7

Answer: C

Note that even if you didn't know how to do this problem, you could eliminate 3 of the 5 answer choices immediately. Because machine 2 can finish the entire job in 3 hours BY ITSELF, then you can see that answer choices A, D, and E are obviously incorrect. When machine 2 gets help from machine 1 to finish the job, they will obviously finish the job in less than 3 hours.

[Brent@GMATPrepNow]

It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?

For this question, I think a good approach is to assign the ENTIRE job a certain number of units.
The least common multiple of 4 and 3 is 12.
So, let's say the ENTIRE production order consists of 12 widgets.

It would take one machine 4 hours to complete a large production...
Rate = output/time
So, this machine's rate = 12/4 = 3 widgets per hour

...and another machine 3 hours to complete the same order.
Rate = units/time
So, this machine's rate = 12/3 = 4 widgets per hour

So, their COMBINED rate = 3 + 4 = 7 widgets per hour.

Working simultaneously at their respective constant rates, to complete the order?
Time = output/rate
= 12/7 hours

Cheers,
Brent

[Matt@VeritasPrep]

The key takeaway here: if the work to be done is *ONE* job, then rate and time are reciprocal.

To see why, consider our equation Work = Rate * Time.

If W = 1, then we have

1 = R * T

So R = 1/T and T = 1/R. That means a rate of 7/12 gives a time of 12/7, and vice versa.