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December 8th, 2016, 03:34 AM   #1
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Something that's confusing me

I have the following question:
If I drive one way at a speed of 40 km/h, what is the speed I should drive back in, in order for the average speed for both ways (back and forth) to be 80 km/h.

After looking at the solution, I understood that it's not possible.
But at first I thought it is simply to drive back at 120 km/h? so the average speed would be 80 km/h? why that's not true?

If you want me to post the solution let me know
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December 8th, 2016, 07:04 AM   #2
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$\overline{spd} = \dfrac{40+x}{2} = 80$

$40+x = 160$

$x=120$

I'd like to know why you think it's not possible to achieve and average speed of 80 km/hr.
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December 8th, 2016, 07:18 AM   #3
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The simple answer is that division by zero is not a valid arithmetical operation.

Remember the basic formula $r * t = d \implies t = \dfrac{d}{r}\ and\ r = \dfrac{d}{t}.$

Let's say that the distance traveled ONE WAY is 160 kilometers. So clearly the distance to and fro is 320.

$Let\ t = total\ time\ going\ and\ coming.$

If the average speed to and fro is 80 kilometers per hour, then

$80 * t = 320 \implies t = \dfrac{320}{80} = 4\ hours.$

Nothing weird so far.

$Let\ u = time\ spent\ going\ and\ v = time\ spent\ coming.$

$\therefore u + v = 4 \implies v = 4 - u.$

With me so far?

But the trip going was done at 40 kilometers per hour for a distance of 160, right?

$\therefore u = \dfrac{160}{40} = 4 \implies v = 4 - 4 = 0.$

Is there any rate of speed that would allow you to travel 160 miles in no time at all? Obviously not. How does this manifest itself mathematically?

The indicated rate on the return trip is

$\dfrac{160}{v} = \dfrac{160}{0},\ which\ is\ not\ a\ real\ number.$

Now you can substitute x and 2x in for 160 and 320 and reach the same result. Division by zero is not a mathematical thing.
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December 10th, 2016, 12:26 AM   #4
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Quote:
Originally Posted by romsek View Post
$\overline{spd} = \dfrac{40+x}{2} = 80$

$40+x = 160$

$x=120$

I'd like to know why you think it's not possible to achieve and average speed of 80 km/hr.
That's what I thought until I saw the answer, as Jeff showed:

Quote:
Originally Posted by JeffM1 View Post
The simple answer is that division by zero is not a valid arithmetical operation.

Remember the basic formula $r * t = d \implies t = \dfrac{d}{r}\ and\ r = \dfrac{d}{t}.$

Let's say that the distance traveled ONE WAY is 160 kilometers. So clearly the distance to and fro is 320.

$Let\ t = total\ time\ going\ and\ coming.$

If the average speed to and fro is 80 kilometers per hour, then

$80 * t = 320 \implies t = \dfrac{320}{80} = 4\ hours.$

Nothing weird so far.

$Let\ u = time\ spent\ going\ and\ v = time\ spent\ coming.$

$\therefore u + v = 4 \implies v = 4 - u.$

With me so far?

But the trip going was done at 40 kilometers per hour for a distance of 160, right?

$\therefore u = \dfrac{160}{40} = 4 \implies v = 4 - 4 = 0.$

Is there any rate of speed that would allow you to travel 160 miles in no time at all? Obviously not. How does this manifest itself mathematically?

The indicated rate on the return trip is

$\dfrac{160}{v} = \dfrac{160}{0},\ which\ is\ not\ a\ real\ number.$

Now you can substitute x and 2x in for 160 and 320 and reach the same result. Division by zero is not a mathematical thing.
Thank you! yet still confusing, as I instinctively did the regular average of 120 km/h , skipped the time part

Last edited by noobinmath; December 10th, 2016 at 12:33 AM.
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December 10th, 2016, 05:20 AM   #5
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Quote:
Originally Posted by romsek View Post
$\overline{spd} = \dfrac{40+x}{2} = 80$

$40+x = 160$

$x=120$

I'd like to know why you think it's not possible to achieve and average speed of 80 km/hr.
Ouch! Surely romsek (not to mention topsquark) knows that taking the average of two numbers is not the same as finding an average speed.

The average speed is the total distance driven divided by the total time taken.
A difficulty here is that the OP does not say what distance is being driven. Let "x" be the distance, in miles, one way. Then the total distance driven is 2x. Since the first leg, distance x, was driven at 40 mph, the time taken was x/40 hours. Let "v" be the speed driving back. Then the time required is x/v so the total time required is x/40+ x/v= x(1/40+ 1/v)= x(v+ 40)/(40v).

The average speed then is 2x/(x(v+ 40)/40v)= 2x(40v/x(v+ 40))= 80v/(v+ 40)= 80. Multiplying both sides by v+ 80, 80v= 80(v+ 40)= 80v+ 3200.

The "80v" terms cancel leaving 3200= 0 which is not true so there is no solution.

Last edited by Country Boy; December 10th, 2016 at 05:23 AM.
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December 10th, 2016, 05:32 AM   #6
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Quote:
Originally Posted by Country Boy View Post
Ouch! Surely romsek (not to mention topsquark) knows that taking the average of two numbers is not the same as finding an average speed.
I plead permanent insanity. I've got the meds to prove it, too!

-Dan
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December 10th, 2016, 08:58 AM   #7
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Quote:
Originally Posted by Country Boy View Post
Ouch! Surely romsek (not to mention topsquark) knows that taking the average of two numbers is not the same as finding an average speed.
yes distance. As in the total ground covered. Not the final distance between the two points.

I sure wish gasoline consumption worked the way you, Jeff, and the problem writer feel average speeds work.
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December 11th, 2016, 04:12 AM   #8
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On average, this was a good thread !
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December 11th, 2016, 12:28 PM   #9
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On average, this was a good thread !
(groans in great pain)

-Dan
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December 11th, 2016, 12:31 PM   #10
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Dan, what's also funny is that I couldn't
stop laughing when I thought of that!
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