Understanding Time and Work

  1. Let M=Men, D=Days, H=Hours, η=efficiency, W=Work

    ##MDH \propto W##

    ##MDH = kW##

    ## \frac {MDH}{W} = k##

    ##\frac {M_1D_1H_1}{W_1} = \frac {M_2D_2H_2}{W_2} = \frac {M_3D_3H_3}{W_3}##

    Which means total total Men hour per unit of work is constant and we say

    η=total work done in one day (or 1 hour)

    My doubts
    1. Does η=W/MDH ?
    2. Does here η refers to Rate of work.
    Last edited by a moderator: Apr 18, 2014
  2. jcsd
  3. Mark44

    Staff: Mentor

    I don't think that efficiency figures in here. Your MDH = kW formula says that if you multiply the number of men by the number of days and the number of hours in a day (wouldn't the number of hours per day be constant?), you get W (or a multiple of W).

    If you want to incorporate efficiency, you are going to have to have a more sophisticated formula.

    For another thing, how do the units match up? On the left you have (men) * (days) * (hours/day), assuming that H represents hours/day and not hours. So the units on the left are man-hours. Are the units of W also in man-hours?
    Last edited: Apr 18, 2014
  4. Considering H=Hours/day still ##\frac {MDH}{W}## refers to total Men hours required to do 1 unit of work. And here Efficiency (η) refers to 1 day work or 1 hour work, then it refers to rate at which work is done. Actually I've seen some people using these words interchangeably but still I'm not sure.
    Last edited: Apr 18, 2014
  5. Mark44

    Staff: Mentor

    No, it doesn't.
    The units of MDH are (men) * (days) * (hours/day), so the units of MDH are men-hours (usually written as man-hours).

    You haven't said what the units of W are in. Possibly it's the number of hours it would take one man to do a job. If that's what W is here, then the units of MDH/W would be (man-hours)/(hours), which would tell you how many men it would take to do a certain job.

    For example, if a certain job would take one man 16 hours, and you have two men, then the job could be done in one day (of 8 hours).
    This makes no sense to me. Your formula with MDH assumes that each worker is working at the same efficiency. In other words, if it would take worker A 8 hours to paint something, then worker B could paint the same thing in the same time.

    As I said before, if you want to incorporate efficiency of each worker into your formula, it's going to have to be quite a bit more complicated. You'll have different expressions for each worker for the amount of work produced in a unit of time.
    Last edited: Apr 18, 2014
  6. @Mark44 Please read the above comment I've edited it.
  7. Mark44

    Staff: Mentor

    This is confusing. At the top, you say η = efficiency. Later on you say η = total work done in one day (or 1 hour), which is really the work rate. If η is efficiency, it would be a number somewhere between 0 and 1.0 (or 0% to 100%). You also haven't defined what "work" means in your formulas. I asked earlier if work represents man-hours, but you haven't answered me.

    Also, your formula would be much simpler if you got rid of "days" and just used "hours".

    A simple example might help clarify things. A general contractor has two painters, Al and Ben. Al can paint a "standard" house in 16 hours. Ben can paint the same sized house in 20 hours. How long would it take both of them working together to paint a standard sized house?

    Al's work rate: 1/16 of a standard house per hour
    Ben's work rate: 1/20 of a standard house per hour

    Let t = number of hours for both working together to paint a standard house.

    The amount of work done is (work rate) * (time), or (houses/hour) * (hours), so
    Al's work done + Ben's work done = 1 house painted

    In symbols,
    1/16 * t + 1/20 * t = 1, or
    (1/16 + 1/20)t = 1
    (5/80 + 4/80)t = 1
    (9/80)t = 1
    t = 80/9 ≈ 8.9
    Working together, Al and Ben can paint a "standard" house in about 8.9 hours.
  8. Don't get confused with η = efficiency that we read in Science and Engineering. Here it is different it means Rate. Follow this link



    Here they have represented η = efficiency as percent
    η(A)=20%=1/5 => 1 day work => which represents rate right
    Yes Work represents man hours
    The reason why I mentioned both, because sometime in question only days is mentioned i.e work done in some days and sometime hours/day is mentioned.
  9. Mark44

    Staff: Mentor

    Like you said, efficiency is well understood in science and engineering. When you use the same term to mean something different, you (or the guy in the Indian Bank Prep web site), you run the risk of confusing people.

    When you set up an equation that represents a physical system, it's important that the units are consistent on both sides of the equation. If you throw around terms such as efficiency and work, and aren't clear on what the underlying units are, then any equation you write will be meaningless. In all of my replies here, I've been trying to get you to think in terms of the units attached to the terms in your formulas.

    Some of the things you wrote are pretty much meaningless, such as this:
    You (and Ramandeep Singh) are clouding the water by using efficiency and the amount of work per unit time synonymously. I agree that 20% = 1/5, but how does that imply "1 day work" and how is "1 day work" a rate?

    A rate is the quotient of two quantities, like distance/time (velocity rate) or mass/volume (density). If you say something is a rate, but don't say what quantities are being divided, you aren't being clear.

    I'm not impressed by the quality of the web site you posted. In the 3rd question, about filling a tank, he writes in his solution (Method 1),
    By what mathematics is 1/3 hour the same as 300%?
    By what reasoning is 60 minutes or 1 hours the same as 100%?
    If "efficiency" here is a rate, what rate (i.e., ratio) are we talking about?

    What he's selling on his web site is a way of solving problems on a banking exam, with absolutely no understanding required.
  10. I think you did't read the question in the picture carefully. In question it is mentioned

    'A' can complete a Job(unit of Work) in 5 days
    5 days -> 1 (1 represents unit of work)
    1 day -> 1/5th of Work this implies in 1 day 1/5th of work is done i.e 1 day work
  11. Mark44

    Staff: Mentor

    No, I did carefully read the question in the image you posted, and I looked at the web page in the link, and read what Mr. Singh had to say for the first three examples. I understand his reasoning for the percentages, but when he writes mathematical equations that are nonsense, I question his ability to explain things.
    The above came from the third example on Singh's page. It's nonsense.
    Last edited: Apr 19, 2014
  12. micromass

    micromass 20,039
    Staff Emeritus
    Science Advisor
    Education Advisor

    This is a pretty horrible site. It might help you to pass the exam, but it doesn't help people understand the solution. In fact, it encourages misunderstanding and memorization. Not my favorite way to do math and science.

    Sure, it will help you reach the right answer. But the right answer isn't the most important thing in science and math (although it might be the most important thing if you're doing the exam).
    1 person likes this.
  13. I agree with you.
  14. Ok lets forget about the term Efficiency (η) for a while. Now does

    ##\frac {MDH}{W} = k##

    represents constant rate.
    Example: Suppose Rates for sets of people doing the same work separately

    Rate for 1st set of people ##\frac {M_1D_1H_1}{W_1}##
    Rate for 2nd set of people ##\frac {M_2D_2H_2}{W_2}##
    Rate for 3rd set of people ##\frac {M_3D_3H_3}{W_3}##

    Now if we consider Efficiency (η) then some people write formula like this

    ##\frac {η MDH}{W} = k##

    What does this represent, if η is considered.
  15. Mark44

    Staff: Mentor

    Rate of what? I keep asking this question in an effort to get you to state things with more precision. A rate represents a quotient, such as miles/hrs (AKA mph) or meters/seconds, and so on.

    Before going further, let's simplify things a bit by getting rid of D in the formula, and just working with hours. So M represents the number of men (all with equal ability) and H represents the number of hours they are all working. With all of them working for H hours, they will accomplish MH man-hours of work.

    If a job requires 60 man-hours of work, then it will take one man 60 hours, or two men 30 hours, or four men 15 hours, and so on. In this formulation MH is proportional to W, with k = 1. More to the point, MH = W.

    Now, let's change things up, and redefine what W means. Suppose the job is to paint a building that has 4000 sq. ft. of surface to be painted.

    So W = 4000 sq. ft. The units of W are area, in sq. ft.

    Suppose Al and Ben are painters, where Al's work rate, RA, is 200 sq. ft./hour, and Ben's work rate, RB, is 250 sq. ft./hour. Notice that the units for the work rates are quotients - sq. ft./hours.

    The amount of work done by either painter is (work rate) * time.

    We can find Al's time to complete the job from this equation: RAtA = 4000, or tA = 20 hours.
    For Ben, the equation is RBtB = 4000, or tB = 16 hours.

    Working together, their rates would add, so to find the time for both of them painting, we have
    (RA + RB)t = 4000, which is pretty easy to solve after putting in their respective work rates.

    Last edited: Apr 23, 2014
  16. Mark44

    Staff: Mentor

    In my previous post I purposely did not say anything about the equations above. My sense is that you are throwing around equations with little or no understanding of what they are supposed to represent. If you look at my responses in this thread, in each one I have put in a lot of effort at explaining what each variable means, and the units involved. When I used the word "rate," I explained in detail what quantity was changing with respect to some other quantity. It doesn't seem to have gotten across, though, when you write about a "rate" with no further description.

    Unless and until you can ask a coherent question, I'm done here.
  17. I'm sorry friends. I got confused because of my teacher, and some bad websites. I checked my notes and my teacher also used the term Efficiency for 'η', I think which is wrong. The correct meaning of η here is rate, As 1 day work/ 1 hour work is a rate.
  18. Mark44

    Staff: Mentor

    "1 day work/1 hour work" isn't a rate. As near as I can make out, this represents the number of hours in a day, which has nothing to do with efficiency or work rates.
  19. @Mark44 See your above comment, You also said the same thing. I'm not saying the individual Work rate is total work done/ day or hour. I'm saying Work Rate = Work done by individual / day or an hour.
    Last edited: Apr 24, 2014
  20. Mark44

    Staff: Mentor

    OK, this is a little closer to being clear. To be more clear, the individual work rate = (work done by the individual)/hour. (Let's get rid of days, and just keep track of time by hours alone. Having days in there as well just confuses the issue.)

    What you wrote before, and what I was commenting on, was this:

    This is not a rate.
    Last edited: Apr 25, 2014
  21. I agree ##\frac {M_1D_1H_1}{W_1}## doesn't represent rate. :wink:
    Last edited by a moderator: Apr 25, 2014
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