159 replies to this topic

[rookie1ja]

Speeding up - Back to the Cool Math Games
If I went halfway to a town 60 km away at the speed of 30 km/hour, how fast do I have to go the rest of the way to have an average speed of 60 km/hour over the entire trip?

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Spoiler for Solution

Speeding up - solution
This one has no solution. Unless we are complicating it with relativity theory - time and space. But to keep it simple, you can't reach the desired average speed under the given circumstances.

However, there is another option for the original wording - a detour. Think about that

Spoiler for old wording

If I go halfway to the town (which is 60 km away) at the speed of 30 km/hour, how fast do I have to go for the rest of the way to have the average speed of the entire way 60 km/hour?

Edit: "rest of the way" means to the town and not an inch farther and the total distance traveled has to be exactly 60 km (this is just to explain how I meant the riddle to be understood)

[Grujah]

And what will my average speed be if I go 90 km/hour for the second half of the trip? Wouldn't it be 60km/hour?

[maryana14]

The two cities are separated by 60 km. If I have traveled half way (30 km) at 30 km/hr then I have spent 1 hour traveling. To reach an average speed of 60 km/hr in the total travel (which is 60 km) I need to spent one hour in total. But I already spent 1 hour in half of the travel. Unless I move instantaneously (infinite velocity) from the half of the travel to the other city, there is no solution.

Traveling at 90 km/hr (the second half) gives an average of 45 km/hr.

[midmathteach]

If you go 90 km/hr for the second half of the trip your average speed would not be 60 km/hr. Here's why:
Average speed = total distance / total time
Your total distance would be 60 km. The first half of the trip takes 1 hr of course, but the second half takes less time since you are traveling faster, it takes only 1/3 hr. And 60km/ (1+1/3 )hr = 45 km/hr.

Another way to explain why it is impossible, is to use algebra:
Let r = rate of second half. So the time for the second half is 30/r.
Again, using the formula:
average speed = total distance / total time
60 = 60 / (1 + 30/r)
60(1+ 30/r) = 60
1 + 30/r = 1
30/r = 0
Which is impossible. (The rate of the second half cannot be 0)

[natterbox]

to get the mean/average you need to add them and divide by 2 to get 60. to do this do it in reverse so you double 60 to get 120 and you know one is 30 so you have 90 left and thats the answer!!!!!

[Veracity]

If you go 90 km/hr for the second half of the trip your average speed would not be 60 km/hr. Here's why:
Average speed = total distance / total time
Your total distance would be 60 km. The first half of the trip takes 1 hr of course, but the second half takes less time since you are traveling faster, it takes only 1/3 hr. And 60km/ (1+1/3 )hr = 45 km/hr.

Another way to explain why it is impossible, is to use algebra:
Let r = rate of second half. So the time for the second half is 30/r.
Again, using the formula:
average speed = total distance / total time
60 = 60 / (1 + 30/r)
60(1+ 30/r) = 60
1 + 30/r = 1
30/r = 0
Which is impossible. (The rate of the second half cannot be 0)

Forgive my ignorance, but I'm still a little confussed.. <--- NOT A MATH TEACHER If you have the ability to speed up the average by doing 90 to make it 45, can you not speed it up more thus raising the average. When it comes to Algebra, I have the mental aptitude of a Lego Block.

Ahhhh, WAIT.....................I GET IT!!! All the Letters and Numbers are a fancy way of saying YOU JUST DO NOT HAVE ENOUGH TIME TO RAISE THE AVERAGE THAT HIGH BECAUSE THE FASTER YOU GO THE LESS TIME YOU HAVE TO ACHEIVE THE DESIRED AVERAGE....
And I was starting to think all that porn I watched instead of doing my homework was a Mistake! Sheeeeeesh. IMAGINE THAT!

[TheOnlyOne]

to get the mean/average you need to add them and divide by 2 to get 60. to do this do it in reverse so you double 60 to get 120 and you know one is 30 so you have 90 left and thats the answer!!!!!

I'm sorry you are wrong....
In order to reach the place at an average speed of 60 km/hour you would need to be able to teleport instantaneously to the place 60 km from the start because you already spent one hour at 30 km just to reach halfway.

[snuff3000]

yes impossible

[larryhl]

Not possible, here's a simpler explanation:

To get speed, divide distance by time => d/t = 60 km/hr
The distance is easy => 60 km
The time to have a speed of 60 km/hr then is also easy => 1 hr

The problem then becomes evident. By going at 30 km/hr for half of the total distance (30 km), you have already used up 1 hour of time, so no matter how fast you go, your total average speed will be less than 60 km/hr. Unless like rookie1ja mentioned, you mess with relativity and get to your destination without having any time elapse =P (i.e. going back in time and then reaching your destination)

[cgrhinc]

Where does it say you only have 2 hour? You can average 60 mph for 10 hours right? There is no reason why you can not drive faster for 1 more hour and have an average of 60mph. Maybe I am reading it wrong.