(Your brother is wrong.Â Well, your brother was close, but the teacher probably wants the exact answer.)

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I can't resist.Â I love explaining these things. Â It is NOT a coincidence that the 3/5 and the 5/3 popped up.Â The problem was carefully designed so that things cancel to make the solution faster.Â This problem is actually easier to solve if you DON'T simplifiy or convert into decimals.Â The way I solve it is to keep track of the units. That way, I don't really even have to think.

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Here is my attempt at explaining:

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Rate needs to end up in miles per hour.Â So, we have 2/3 mile, and we have 2/5 hour.Â Because we want to end up with miles per hour, we probably want to divide the thing with the mile, by the thing with the hour. So that is why you need to take the 2/3 and divide by the 2/5.

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/3) mileÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/3)Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â 2 * 5Â Â Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â Â Â **5Â Â Â Â mile**

Rate =Â --------------------------- =Â --------------Â Â Â Â Â -------Â = ------------Â Â Â -------- =Â Â Â Â Â **-----Â -----**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/5) hrÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/5)Â Â Â Â Â Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â 3 * 2Â Â Â Â Â Â Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â Â **3Â Â Â Â Â hr**

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Great!Â The rate ended up in mph!

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Next, to figure out the time.

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Well, time needs to end up in hours.Â We've got something in miles per hour (rate), and we've got something in miles (distance).Â So if we take the something in miles, and divide it by the something in miles per hour, we will get hours.Â So that is why it makes sense to use

time = distance / rate:

Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â distanceÂ Â Â Â Â Â Â Â 1 Â Â Â mile Â Â Â Â Â Â Â Â Â Â Â Â Â (1 mile) (3 hr)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **3**

time = --------------Â =Â Â Â -----------------------Â Â Â = -----------------------Â Â Â =Â Â Â **------Â Â hr**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â rateÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â (5 mile)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **5**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â -----Â Â Â Â --------

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â Â Â Â Â Â hr

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Â

But because we think in minutes, instead of in hours, you now have the question:Â How many minutes isÂ time = 3/5 hr ?Â

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Well, we know that 60 minute = 1 hour, so you can just take your time in hours (the 3/5 hr), and multiply the numerator and denominator by the same time interval:

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â Â 60 minÂ Â Â Â Â Â Â Â Â Â Â Â 3 x 12 Â hr x min

time =Â Â Â Â Â Â ------------- x Â ---------------Â =Â Â Â Â ------------ ------------ Â = **36 minutes**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1 hrÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1 x 1Â Â Â Â Â Â Â Â Â hr

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BTW, to figure out the rate (rate = 5/3 mph) in decimals, you just take the 5 and divide it by 3, and you get **rate = 1.67 mph**.Â

Â Â Â Â Â Â Â Â Â But I assume the teacher would prefer to be given the answer in the form of **rate =Â 5/3 mph**.

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I apologize if this is as clear as mud, but I promise you that it is 100% correct.Â

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By now, this answer probably comes to you too late, but just think how smart you'll sound when your dd gets home from school this afternoon!Â

Edited by emilysmama - 11/8/13 at 10:53am
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