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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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• 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.
• 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?
• 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 !
• 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.
• 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.
• why do i have to take the inverse?
• 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)
logically how would two people working together take more time than working indivisually?
• 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
• 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! :)
• why I get different answer (wrong) when using A(deck/hour) not (1/a)(hour/deck)?
• 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.
• 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)
• 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?
• 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).
• 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.