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Rational equations word problem: combined rates (example 2)

Sal solves a word problem about the combined deck-staining rates of Anya and Bill, by creating a rational equation that models the situation. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Josh
    Why must the rate problem always be set up in deck/hr to get the correct answer.
    i.e. Why can't you put 8hrs/deck=A hrs/deck + B hrs/deck
    If you solve this way using 2A=B you get an incorrect answer but I don't understand why, this was how I tried to solve it originally.
    I'm just confused about how to pick where the A and B will go (on the numerator or denominator) and how to set up the problem.
    Please help!
    (51 votes)
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    • leafers seedling style avatar for user Ezequiel Santana
      CasualJames, I'm with you, but why is the answer wrong? In real life seems logical: you can take 8 hours to run a mile or you can run a mile in 8 hours. But mathematically it makes no sense. That is: 1/2 + 1/2 = 1 yet 2+2=4 so you obviously cannot shift the denominators as you will. But the question remains: Is there a bullet proof method to decide which one is the right denominator or is it mere intuition?
      (21 votes)
  • blobby green style avatar for user Harikesh
    Another way of looking into this problem

    Divide porch into 3 Parts. At the end of 8 hours Anaya would paint 2/3 of porch and Bill would Paint 1/3 of porch. (because bill is twice as slow as Anaya OR Anaya is twice as fast as Bill).
    Conclusion 1 : Anaya takes 8 hrs to paint 2/3 rd of a porch
    Conclusion 2 : Bill takes 8 hrs to paint 1/3 rd of porch.

    Now it boils down to ratios and proportions problem

    Time taken by Anaya to Paint 1 Porch can be given by

    2/3 = 8 then 1 = X

    2/3 = 8
    3/2 *2/3 = 8 * (3/2) ............................... (Multiply both side by 3/2)
    1 = 12

    Similarly Time taken by Bill to Paint 1 Porch can

    1/3 = 8
    1/3 * 3 = 8 * 3....................(Multiply both side by 3)
    1 = 24

    Hence Anaya takes 12 hrs to paint 1 Porch and Bill takes 24 hrs to Paint 1 Porch.

    Hope that helps !
    (17 votes)
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  • leaf grey style avatar for user smatlaldo
    Anyone else who has no problems with calculus but is trying to do mixtures and combined rates for like 3 days now with no success.
    (11 votes)
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  • leaf blue style avatar for user james
    why do i have to take the inverse?
    (5 votes)
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    • leafers tree style avatar for user prachi.ak.26
      because if we didnt inverse it, it would be like this

      8 = A + B
      B = 2A

      8= A + 2A
      8= 3A
      A = 8/3
      A = 2.66 (and repeating)
      B = 5.33(and repeating)
      now think about it.
      logically how would two people working together take more time than working indivisually?
      (7 votes)
  • male robot hal style avatar for user John Eagle
    Why does the following not produce correct results?
    a+b=8
    a+2a=8
    3a=8
    a=8/3=2.66
    b=16/3=5.33
    (3 votes)
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    • boggle purple style avatar for user lily J
      Anya and Bill are working TOGETHER. So they need to finish at the same time. Because Anya could working faster, she could do more than bill does. So why should Anya only use 2.66 hours to finish and just sit there waiting for Bill to finish his part?

      Hope that helps! :)
      (4 votes)
  • blobby green style avatar for user Hiro
    why I get different answer (wrong) when using A(deck/hour) not (1/a)(hour/deck)?
    (4 votes)
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    • duskpin ultimate style avatar for user TotallyNotAFurryÒwÓ
      When you use A (deck/hour) instead of 1/A (hour/deck), the units won't cancel out correctly in the equation, which will lead to a different answer that is not correct.

      Let's see why:

      If we use A (deck/hour) instead of 1/A (hour/deck), the equation will become:

      8 = A + B

      where A represents the rate of Anya in decks per hour, and B represents the rate of Bill in decks per hour.

      But we know that the rate is equal to the inverse of the time, which means that A is equal to 1/x (hour/deck) and B is equal to 1/y (hour/deck), where x is the number of hours Anya takes to paint a deck, and y is the number of hours Bill takes to paint a deck.

      Substituting 1/x and 1/y for A and B in the equation, we get:

      8 = 1/x + 1/y

      If we solve this equation for x and y, we will get different values than the correct answer.

      Therefore, it's important to use the correct units when defining the variables and writing the equation to ensure that the units cancel out correctly and we get the correct answer.
      (2 votes)
  • spunky sam blue style avatar for user Akbar Khuwaja
    At , why does Sal decide to multiply "1/8 = 1/A +1/2A" by 8A? I am wondering why does it answer the question correctly? I ask because I added "1/A + 1/2A" to come up with "1/8 = 1/3A." Then I inversed it to get "8 = 3A" which is "A = 8/3" or "A = 2 2/3 hours per deck (or 2 hours and 40 minutes). While I came up with 2 hours and 40 minutes for Anya, Sal came up with 12 hours for her. Somebody please help me understand why is my calculation incorrect? Thank you.
    (1 vote)
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  • blobby green style avatar for user Arbaaz Ibrahim
    If two people work on a task at the same rate, what's the reasoning behind, the task being done in half the amount of time?
    Is it something calculated mathematically, or is there some logical reasoning?
    I understand that, it will be quicker, but what's the reasoning behind it being half the amount of time, than the times of the individuals?
    (2 votes)
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  • leaf red style avatar for user Arkan Sharif
    There are no exercises, what do I now?
    (1 vote)
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  • blobby purple style avatar for user DhruvaSB
    why is it when I try to solve it like this:
    a+b=1/8 ----> where a and be are in decks per hour
    2a-b= 0 ----> cause Bill take 2 times longer than Anya
    that I get a wrong answer?
    (2 votes)
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    • blobby green style avatar for user fatman321
      The rate at which Anya paints a deck is 1/A decks per hour. Similarly, the rate at which Bill paints a deck is 1/B decks per hour. A and B represent the number of hours it takes Anya and Bill, respectively, to paint the deck by themselves, not the number of decks they can paint in 1 hour. Therefore, the sum of their individual rates (in decks per hour) would not be A + B, but 1/A + 1/B, which would equal 1/8. Try solving the problem using 1/A + 1/B rather than A + B in your first equation.
      (2 votes)

Video transcript

Working together Anya and Bill stained a large porch deck in 8 hours. Last year Anya stained the deck by herself. The year before Bill painted it by himself, but took twice as long as Anya did. How long did Anya and Bill take when each was painting the deck alone? So let's define some variables here. Let's define A. Let's define A as number of hours for Anya to paint a deck. And so we could say that Anya paints, or has A hours for one deck. Or we can invert this and say that she can do 1/A decks per hour. Now, let's do another variable for Bill just like that. Let's define B as the number of hours for Bill to paint a deck. And so for Bill, he can paint-- it takes him B hours per deck, per one deck. We could put it over 1 there if we want. We don't have to. And this is the same thing as saying that he can do 1 deck per every B hours. Or another way to think of it, is he can do 1/B deck per hour. Now, when they work together, Anya and Bill stained a large porch deck in 8 hours. So let's write this down. Anya plus Bill-- I'll do it in orange. Anya and Bill. This information right here. Anya and Bill stained a large porch deck in 8 hours. So we could say 8 hours per deck. Or 8 hours per 1 deck, which is the same thing as saying 1 deck per 8 hours. And this is going to be the combination of each of their rates. So this 1 deck of per 8 hours, this is going to be equal to Anya's rate, 1/A decks per hour, plus Bill's rate. Plus-- do that same color. Plus 1/B decks per hour. So we have one equation set up. Let me scroll down a little bit. And I won't write the units any more. We have 1/8 is equal to 1/A plus 1/B. Now, we have two unknowns, so we need another equation if we're going to solve these. Let's see. It tells us, it tell this right here the year before Bill painted it by himself, but took twice as long as Anya did. So the number of hours it takes Bill to paint the deck is twice as long as the number of hours it takes Anya to paint a deck. So B is equal to 2A. The number of hours Bill takes is twice as the number of hours Anya takes per deck. So B is equal to 2A. So we can rewrite this equation as-- I'll stick to the colors for now. We could say 1/8 is equal to 1 over Anya. Instead of writing 1 over Bill we would write, so plus 1 over Bill is 2 times Anya. The number of hours Bill takes is two times the number of hours Anya takes. So 2 times Anya. And now we have one equation and one unknown. And we can solve for A. And the easiest way to solve for A right here is if we just multiply both sides of the equation by 2A. So let's multiply both sides of this by 2A. Multiply the left side by 2A. Actually, let's multiply both sides by 8A, so we get rid of this 1/8 as well. And then multiply the right-hand side by 8A as well. The left-hand side, 8A divided by 8 is just A. A is equal to-- 1/A times 8A is just going to be 8. 1/2A times 8A. 8A divided by 2A is 4. So A, the number of hours it takes Anya to paint a deck-- and I made it lowercase, which I shouldn't have. Well, these should all be capital A's. This is A is equal to 8A over A is 8. 8A over 2A is 4. So the number of hours it takes Anya to paint a deck, or A, is 12 hours. Now what are they asking over here? They're asking, how long did Anya and Bill take when each was painting alone? So we figured out Anya. It takes her 8 hours. And then we know that it takes Bill twice as long as Anya. Bill is two times A. So bill is going to be 2 times 12. So Bill is equal to 2 times Anya, which is equal to 2 times 12 hours. Which is equal to 24 hours. So when they're alone it would take Anya 12 hours, it would take Bill 24 hours. When they do it combined, it takes 8 hours. Which makes sense. Because if Bill took 12 hours by himself, combined they would take 6 hours, they would take half as long. But Bill isn't that efficient. He's not as efficient as Anya, so it takes them a little bit longer. Takes them 8 hours. But it makes sense that combined they're going to take less time than individually.