If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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. • 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?
• why do i have to take the inverse? • 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 ! • 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 • 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? • There are no exercises, what do I now?
(1 vote) • Working together, it takes two different sized hoses
35
minutes to fill a small swimming pool. If it takes
55
minutes for the larger hose to fill the swimming pool by itself, how long will it take the smaller hose to fill the pool on its own?
(1 vote) • Because the hoses are different we need to consider how much the pool is filled per minute.
For the larger hose 1/55 of the pool will be filled in a minute. The smaller hose would then be 1/x per minute. Now we can add both hoses together (1/x + 1/55) and set it equal to how much of the pool is filled when both are working (1/35)
1/x +1/55 = 1/35 and solve
1/x=1/35-1/55
1/x=4/385
4x=385
x=96.25
So it would take 96.25 minutes to fill the pool with just the smaller hose.
• How can we prove that both of them together take 8 hours to paint?
(1 vote) • why multiply both sides of the equation with 8A
I think we can figure out the answer without multiply 8A.
1/8= 1/A + 1/2A
1/8= 3/2A
2A=24
A=12
I'm confusing that how did 8A come from.
(1 vote) • Your method does work. You are working with the fractions and then using properties of proportions to finish.
Sal's method is good if you don't want to do a lot of math with fractions. The 8A is the LCM for all the denominators in the equation. If you multiply the equation by 8A, the denominators cancel out and you have an equation with no fractions. Some people find this easier to do than your approach. So, it's good to have options.