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### Course: College Algebra > Unit 8

Lesson 2: Modeling with rational expressions# 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.

## Want to join the conversation?

- 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)- 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)

- 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.(22 votes)
- I can't post a link, because KhanAcademy only allows links to their pages. But, if you do an internet search for "mixture problems bucket method" at mgccc.edu that might help you.(9 votes)

**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)- why do i have to take the inverse?(6 votes)
- 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)

- 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)- 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)

- Even if you are a master of mathematics this is extremely tough! The best advice is to write down the steps to solve.

1. Identify the strategy and formula's you need as well as the logic to derive the right formula. These are not given so you will be very confused, just get hints and review the answers.

2. Write down the steps you need to solve this in order from 1-7 detailed below. By writing it down in a sequence, it will help break down the complexity of the problem and help you solve it piece by piece instead of being overwhelmed and completely lost.

For example on mixtures solving for volume:

1. Identify strategy/formulas

2. Solve for total volume (equation)

3. Solve for ingredient volume

4. Solve for volume of secondary variable you must solve for

5. Combine and derive equation

6. Solve for the mixture composition using what you obtained from #5

7. Use this same numbered step format to identify how to solve each part in proper order. By breaking it down into parts like I just did above it will make it more comprehensible. Do not worry if you do not fully comprehend this, aim for 75% mastery during the first 3 days and over time you will start to understand better. Don't stress out, do your best and solve the easier problems at first and learn the format. You can obtain full mastery in about 2-3 days with rest if you are a master of previous lessons.

You may want to skip this and complete the rest of the course and then revisit this at the end.(5 votes) - why I get different answer (wrong) when using A(deck/hour) not (1/a)(hour/deck)?(4 votes)
- 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.(1 vote)

- 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)- It is mathematical. If 2 people are working together on one task, and they work at the same rate of speed, then each person completes 1/2 the work needed to complete the job and the whole job gets finished because 1/2+1/2 = 1 (the job is done).(4 votes)

- At4:06, 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)
- you can't just add denominators (a+2a), you have to make them equal (multiply 1/a times 2 to make it equal to second fraction):

1/a + 1/2a

2*1/2*a + 1/2a

2/2a + 1/2a = 3/2a

more info here: http://www.khanacademy.org/math/arithmetic/fractions/v/adding-and-subtracting-fractions(4 votes)

- There are no exercises, what do I now?(1 vote)
- If KA has no exercises for this topic, you can search the internet for other sites that do.(4 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.