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Rational equations word problem: combined rates

Sal solves a word problem about the combined leaf-bagging rates of Ian and Kyandre, 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
    I have the gut feeling to start out with their rates in hours/lawn. Thus their combined rate (K + I) would be 3hours/lawn + 5hours/lawn= t/lawn or total hours per lawn. But this is obviously incorrect. What am I doing wrong? Sal starts out correctly with lawn/hour. Why is my intuition to start out with hour/lawn instead of lawn/hour is wrong. I just can't figure out why one rate (hour/lawn) is incorrect to solve the problem while going with another rate (lawn/hour) is correct.
    (45 votes)
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    • leaf green style avatar for user ksinelli
      Hours per lawn is not an unreasonable intuition to start off with, but we can quickly see that adding their times together to get 8 hours per lawn doesn't make sense, as the time should get shorter when they are working together.

      What you really want to ask yourself is "How much can each person do in a given time?" When the amount of time is the same for both people, we can get a much better comparison of their efforts.
      This is why Sal talks in terms of one third of a lawn per hour, and one fifth of a lawn per hour.
      It is in this way that you can add their efforts together to see how much of one lawn they can complete in one hour.

      That's the way I see it at least. Hope it helps.
      (44 votes)
  • blobby green style avatar for user Harikesh
    It is not so intuitive for me.
    Say Ian takes 5 hrs for 1 lawn & Kyandre takes 3 hrs for 1 lawn.
    So they take 8 hours for 2 lawns and hence 4 hours for a lawn ?
    what is wrong in above assumption ?
    Please help !
    (4 votes)
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    • piceratops ultimate style avatar for user abassan
      They don't take 8 hours for 2 lawn.
      Here is why. After 3 hours Kyandre has finished 1 lawn and Ian has finished 3/5 of his lawn. At this point, Kyandre, helps Ian with the second lawn and they finish faster than Ian would alone.
      Does that help you understand what is wrong with the assumption?
      (30 votes)
  • male robot johnny style avatar for user Hans Liu
    At , why can Sal flip the left and right sides and just use the reciprocal of both sides? I can't remember the rule or property that justifies it. Help would be greatly appreciated, thanks. Also, I tried to do it by cross multiplying but couldn't figure out how the units get changed from lawns per hour to hours per lawn.
    (8 votes)
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    • leaf green style avatar for user Lucas Van Meter
      One way to see it is that Sal is doing a few steps at once. He starts with 8/15=1 (lawn)/t (hours). Next lets multiply both sides by t (hours) to cancel the t (hours) on the right and get (8/15)t (hours)=1 (lawn). Next multiply both sides by 15/8 to cancel the 8/15 on the left and get t (hours)=15/8 (lawn). As a last step we could divide both sides by 1(lawn) to get t (hours/lawn)=15/8. That is how he did it.

      You could also justify taking the reciprocal by the following reasoning. If A=B then 1/A=1/B (assuming A and B are not 0). This is true since we can directly substitute B in for A because of the equality. In general if you have an equality you can do whatever you want to both sides as long as you do it to both and you don't do something like dividing by 0.

      As a side note, a real example of reciprocal units is period and frequency of waves. Period measures seconds per cycle and frequency measures cycles per second. If the frequency is 30 beats per minute than the period is 2 seconds per beat.
      (12 votes)
  • ohnoes default style avatar for user garrettvilloria
    at , why does Sal multiply 60 by 7/8? What happened to the 1 7/8?
    (2 votes)
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    • aqualine tree style avatar for user Judith Gibson
      The time needed if they both worked together (t) was 1 7/8 HOURS.
      So Sal kept the 1 hour separate and then converted 7/8 of an hour to minutes by multiplying by 60 min/hr.
      After he got the answer of 52.5 minutes, he then combined that with the 1 hour to get the final answer of 1 hour 52.5 min.
      (11 votes)
  • starky ultimate style avatar for user Averatu
    The product over sum formula is one of the most useful tools in applied math imho.
    (3*5)/(3+5) = 15/8
    (5 votes)
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  • starky ultimate style avatar for user AkamantoMorpheus2
    By now, i'm quite familiar with work rate problems, but I have hit a particular snag that no one on the internet seems to have addressed...
    How do I algebraically represent one of the two workers leaving the job before it's complete? Here is the problem:

    " A man can lay a concrete sidewalk in 5 days; his assistant can lay the same sidewalk in 8 days. After working together for 2 days, the man is called away. How long will it take the assistant to finish the work? "

    I have gotten so far as to determine that it would take the two workers 40/13 days to lay the sidewalk if they were to work together the whole way through, but after that, I'm completely stuck. I'm sure i'm not the only one who has seen a problem like this, but as i have scoured the internet for examples of such problems, it seems that either no one else has seen such a problem, or I'm missing something painfully obvious. can somebody clue me in?
    (2 votes)
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    • leaf blue style avatar for user Sukhov
      In order to treat such problems more easily you first need to convert (in mind or on paper) the rates into (y/x part of the work done / (per) time interval notation) which in your particular problem would be 13/40 (combined efforts) / (per) day. It may seem not that intuitive but 13/40 represents the same daily rate but in a more disguised way.
      Personally I find very confusing multiplying 13/40 (sidewalks/days) by 2 days (different frame of reference) but I have no problem when I know that per day 13/40 of sidewalk is done and when 2 days pass I just double the rate.
      Thence, when you start solving a problem it is much more keep track of what you are doing when you keep in mind the same frame of reference.

      I would apply similar thought process to the second part.

      Cheers.
      (3 votes)
  • piceratops ultimate style avatar for user Mark
    When combining the rates my intuition tells me to take the average of the 2 rates (8/30) rather than summing them (8/15). Can anyone explain the logic as to why you sum them? I cant quite get my head around it.
    (2 votes)
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    • blobby green style avatar for user nenadmilosjr
      Consider the case in which one person can rake 1 lawn per hour. It would take one hour to rake one lawn, but if 2 people working at that rate raked the lawn, it would take them half as much time. As a general rule if you want to get the intuition behind a problem try solving the easiest possible variation.
      (2 votes)
  • duskpin ultimate style avatar for user algebeautiful
    How would you do this if it was reversed? For example, Fred can rake a lawn in 5 hours. With Johns help, it takes 2 hours. How long does it take John to rake a lawn?
    (2 votes)
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  • leaf green style avatar for user anonymous user
    I've always wondered, what program does he use for these videos? Is it a custom thing? Is there some way I could download it? (not that I particularly 𝘸𝘢𝘯𝘵 to download it, I just wonder)
    (2 votes)
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  • orange juice squid orange style avatar for user annie
    I was searching this topic on another website, and they used the strategy of this: Let a and b be the two different times. So in this case with the lawns, a=5, and b=3. You can do a times b/ a+b. So with the lawns, 5 times 3 is 15, and 5+3 is eight. So you still get 15 over eight, but I also saw many websites using the strategy you used, and I see it takes more time. So which way is more efficient? (I hope you can understand the equation, don't know what letter to use for multiply.)
    (2 votes)
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Video transcript

Ian can rake a lawn and bag the leaves in 5 hours. Kyandre can rake the same lawn and bag the leaves in 3 hours. Working together, how long would it take them to rake the lawn and bag the leaves? So let's think about it a little bit. It says Ian can rake a lawn and bag of the leaves in 5 hours. So for Ian, I for Ian. For 1 lawn, he can do 1 lawn in 5 hours. We could have also written this as 5 hours per lawn. But we'll see that writing it this way is more useful, because it's actually a rate. Because this is the same thing as 1/5 lawns per hour. Or 1/5 of a lawn per hour. That's the rate at which Ian can rake a lawn, at 1/5 of a lawn per hour. Now let's do the same thing for Kyandre. And this actually probably shouldn't be plural. 1/5 lawns, 1/5 of a lawn. So let me just erase that S right there-- 1/5 of a lawn per hour. Now let's do the same thing for Kyandre. Kyandre can rake the same lawn and bag the leaves in 3 hours. So for Kyandre, Kyandre can, for 1 lawn-- I'll assume it's a boy's name-- he can do it in 3 hours. Or if we were to write it as a rate, this is 1/3 of a lawn per hour. Now let's think about what the combined rate is. So let's say if we have Ian plus Kyandre. What's going to be their combined rate? Well, they tell us that working together, how long would it take them to rake and bag the leaves? So let's let let's let t be how long it will take them together. So that's how long they would take together. And if we say t is how long they take together, then we could say that combined, they will do 1 lawn in every t hours, if we're assuming t is in hours. For every t hours. Or as a rate, their combined rate is going to be 1/t of a lawn per hour. So that's their combined rate. So here, we have the rate of Ian and the rate of Kyandre, and the combined rate. So the combined rate's just going to be the sum of each of their rates. If he can do 1/5 of a lawn per hour and he can do 1/3 of a lawn per hour, their combined rate is going to be 1/5 of a lawn per hour plus 1/3 of an hour. Because in an hour he'll do 1/5, and he'll do 1/3. So you'll add those two together to figure out how much they can do in an hour. So their combined rate is going to be 1/5 lawn per hour. And I won't write the units here, just because it gets redundant. 1/5 lawn per hour I could write over here. Plus 1/3 lawn per hour for Kyandre. That's going to be their total rate, which is 1/t lawns per hour. And now we just have to solve for t. And we'll know the total number of hours it will take them. So let's do that. So to do that, we just have to add 1/5 plus 1/3. Well, we have a common denominator of 15. So this is the same thing as 3/15 plus 5/15 is going to be equal to 1/t. And then we have a common denominator now. So 3 plus 5 is 8. So this is going to be 8/15. I'll go over here now. So now we have 3 plus 5 is 8. Over 15 is equal to 1 over t. If we want to solve for t, we could take the reciprocal of both sides. So if we flip the left side, we get 15/8. And if we flip the right side, we get t/1, or just t. So it'll take them 15/8 hours. Or if we want that in kind of a way that we can think about it a little bit better, 15/8-- so t is equal to 15 over 8 hours. And I should say 15 over 8 hours per lawn. This whole time here we had lawns per hour. This was lawn per hour. And this was here as well, lawns per hour. When we flip it, it becomes hours per lawn. So that's exactly what we want. But 15/8 is the same thing as 1 and 7/8 hours. And 7/8 of an hour-- we can get our calculator out. If we have 60 minutes in an hour, times 7/8, we get 52.5 minutes. So this is equal to our answer. Combined, it will take them 1 hour and 52.5 minutes per lawn. Or to do this lawn right over here, the lawn in question. Hopefully, you found that useful.