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September 15th, 2010, 10:08 AM   #1
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Word Problem question, please help.

How do you solve this and rather quickly:

Jennifer paints the barn alone in 3 hours and Mike in 6 hours. How long does it take them to do the work together?
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September 15th, 2010, 10:46 AM   #2
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Re: Word Problem question, please help.

Well, the quick and easy way to solve a problem of this type is to take the product of the times divided by the sum of the times to find how long it takes.

t = [(3 hr.)(6 hr.)]/[(3 + 6)hr.] = 2 hr.

Another way to look at it, based on a recent post by [color=#00BF00]skipjack[/color], is that in 6 hours the two working together could paint 3 barns, or 1 barn every two hours.
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September 15th, 2010, 11:50 AM   #3
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Re: Word Problem question, please help.

This does tend to be one of those problems that certain people excel at on SAT tests, but to further explain the logic so that you can do more complicated problems......

Rate * time = # barns painted

so Jennifer's rate = 1 barns / 3 hours = 1/3 barns/hour
and Mike's rate = 1 barns / 6 hours = 1/6 barns/hour

the combined rate = 1/3 barns/hour + 1/6 barns/hour = 3/6 barns/hour = 1/2 barns/hour
and so
combined time = (# of barns) / rate = 1 barn / (1/2 barns/hours) = 2 hours
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September 15th, 2010, 01:05 PM   #4
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Re: Word Problem question, please help.

One way to derive the product over sum formula is to say that if entity 1 can complete a task in x units of time and entity 2 can complete the same task in y units of time, and working together it takes the two entities T units of time. We have after one unit of time 1/T of the task completed, where entity 1 has done 1/x of the task and entity 2 has done 1/y of the task and their efforts add together for the total. Mathematically, this is stated:

1/T = 1/x + 1/y = (x + y)/(xy)

Solve for T:

T = (xy)/(x + y)
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September 15th, 2010, 01:31 PM   #5
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Re: Word Problem question, please help.

Thank you MarkFL and followingmycoffeecup. The formula of the problem is very helpful on the SAT since one does not have the time to go through all the middle steps.

Do you also have a formula for this distance, rate and time problem at Youtube:

http://www.youtube.com/watch?v=KkF7qq0glLA
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September 15th, 2010, 02:05 PM   #6
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Re: Word Problem question, please help.

Suppose a slower moving entity begins ?t units of time along a shared path before a faster moving second entity. How long will it take for the faster moving entity to catch the slower one? Let the slower moving entity have a constant velocity s and the faster moving entity have a constant velocity f. Since when the faster entity catches the slower, they both will have traveled the same distance d, we have from d = rt:

s?(t + ?t) = d = f?t
st + s?t = ft
s?t = ft - st = t(f - s)

t = (s??t)/(f - s)

In the case of the Pandas, this would give [(45 ft/min.)?(2 min.)]/[(65 - 45)ft./min] = (90/20) min. = 9/2 or 4.5 min.
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September 16th, 2010, 05:14 PM   #7
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Re: Word Problem question, please help.

MarkFL, thanks again.

Do you think students should remember those formulas? In the distance problems, there are quite a lot variations of the above problem. What do you think the best way to approach those problems?
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September 16th, 2010, 05:25 PM   #8
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Re: Word Problem question, please help.

If I were a teacher, I would not want my students to merely memorize formulas, I would rather them understand the concepts behind the formula, and be able to derive them as needed. However, if time is an issue, memorization can certainly speed things up, so it really depends upon the setting.

Some formulas, such as the quadratic formula, are very useful to have memorized, but if a student cannot derive this formula by completing the square for the general quadratic, then something is missed by its usage, in my opinion.

My freshman physics professor, in a calculus-based course, required us to derive all formulas before we were allowed to apply them on our exams. I felt this was a very effective way to ensure we actually understood the material, rather than regurgitating memorized formulas to plug data into.
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September 17th, 2010, 06:24 PM   #9
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Re: Word Problem question, please help.

Thanks for your suggestions. Both of you are very knowledgeable in terms of general approaches, because you can teach hundreds of problems, students may still not get it. At the site where that Youtube video originated, there is a teaching type of article with some interesting research references. I was wondering if anyone can share thoughts on their approaches to those tough word problems. Would that be effective? or is there some better way to learn and teach?

The article is at:

<link removed by moderator due to technical issues with it>

Thanks.
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September 17th, 2010, 08:13 PM   #10
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Re: Word Problem question, please help.

The way I was taught to approach word problems is to first identify the unknown, what it is that needs to be found, and then to identify how this unknown relates to what is given, not all of which will necessarily be useful. Like any skill, it just takes practice. I know this doesn't help much, but like Euclid reportedly said to King Ptolemy, "There is no royal road to geometry."
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