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### Course: Integrated math 3 > Unit 13

Lesson 5: Modeling with rational functions- Analyzing structure word problem: pet store (1 of 2)
- Analyzing structure word problem: pet store (2 of 2)
- Rational equations word problem: combined rates
- Rational equations word problem: combined rates (example 2)
- Rational equations word problem: eliminating solutions
- Reasoning about unknown variables
- Reasoning about unknown variables: divisibility
- Structure in rational expression

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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.

## Want to join the conversation?

- 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)
- 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.(47 votes)

- 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 !(5 votes)- 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?(33 votes)

- At4:08, 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.(7 votes)
- 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.(13 votes)

- at4:59, why does Sal multiply 60 by 7/8? What happened to the 1 7/8?(2 votes)
- 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)

- What is the formula for this?(3 votes)
- Not every problem should have a formula.(7 votes)

- The product over sum formula is one of the most useful tools in applied math imho.

(3*5)/(3+5) = 15/8(6 votes) - 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.(4 votes)
- 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.(4 votes)

- 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?(1 vote)- 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)

- so what was the answer?(2 votes)
- one hour and 52.5 minutes per lawn(2 votes)

- 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)
- You would set the problem up as:

1/5 + 1/x = 1/2

Fred does 1/5 in 1 hour

John does 1/x in 1 hour, where "x" is how long it takes him to do the entire lawn.

Their combined rate = 1/2 done in 1 hour.

Solve for "x" to find John's rate.

Hope this helps.(1 vote)

## 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.