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# Speeding up

159 replies to this topic

### #31 megvilleus

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Posted 10 July 2007 - 07:24 PM

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?

The solution of not possible makes sense to me, and I understand the questions of the people above. I am going to try to answer those questions, but I'm bad at wording, so sorry if it doesn't help.

The question is asking for the average speed of the ENTIRE trip. Therefore, with a total distance of 120 km and an average speed of 60 km/hour, the average time you must take is 2 hours. Because you already took 2 hours to complete the first half, you are out of time.

Now the question is why can't you speed up to 120 km/hour for the second half and then average that to get 60 km/hour. Well, the thing is that when you average the two speeds to get the average speed for the entire trip, you come up with (30 + 120)/2=75 km/hour. As you can see, this is not 60 km/hour and does not satisfy the problem. You must be wondering why I didn't include the distances in that small equation. Well, it is asking for average SPEED, not average speed OVER A GIVEN DISTANCE. Because of that, you can only use the two speeds you went at to average 60.

I apologize for the last couple of sentences, because I kind of repeated myself. I hope that makes sense.
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### #32 savagegamer90

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Posted 10 July 2007 - 07:37 PM

Now the question is why can't you speed up to 120 km/hour for the second half and then average that to get 60 km/hour. Well, the thing is that when you average the two speeds to get the average speed for the entire trip, you come up with (30 + 120)/2=75 km/hour. As you can see, this is not 60 km/hour and does not satisfy the problem. You must be wondering why I didn't include the distances in that small equation. Well, it is asking for average SPEED, not average speed OVER A GIVEN DISTANCE. Because of that, you can only use the two speeds you went at to average 60.

I don't agree with this statement for two reasons.

1. if you went an average of 75 km/hr and only traveled 60km, you would be there in less than an hour, which is imposible, because it took you an hour just to get halfway (going 30km/hr).

2. using this equation you could go at 90km/hr (30 + 90)/2 = 60 and end up with an average speed of 60, which answers the original question.
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### #33 Incognitum

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Posted 10 July 2007 - 08:07 PM

I don't agree with this statement for two reasons.

1. if you went an average of 75 km/hr and only traveled 120km, you would be there in less than an hour, which is imposible, because it took you two hours just to get halfway (going 30km/hr).

??? traveling at 75km/h it takes almost 2 hours to travel 120km, but the formula dictates that he travels 150 miles, which takes exactly 2 hours.

2. using this equation you could go at 90km/hr (30 + 90)/2 = 60 and end up with an average speed of 60, which answers the original question.

What's wrong with that? This is the solution that has been advocated by several people, including myself.
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### #34 savagegamer90

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Posted 10 July 2007 - 09:17 PM

sorry I meant only 60km, not 120km (it has been edited), and on the first page the original question is 60km, I don't know where you got the 150 km from.

Average speed is defined as the distance traveled divided by the time taken.

You can't just add two speeds together and divide them by 2 to get an average, and you can't just ignore distance either.

If you have taken 1 hour to go halfway(30km/hr), and you want to be there in an hour(average 60km/hr), the only way to do that is to move from halfway to finish (30km)instantaneously, which we can ignore as a solution for this problem.
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### #35 Incognitum

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Posted 10 July 2007 - 09:25 PM

sorry I meant less than two hours(it has been edited), and on the first page the original question is 120km

Okay, 2 hours makes sense, but the original question is 60km, not 120.
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### #36 smheath

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Posted 11 July 2007 - 04:48 AM

For those saying the answer is 90 km/h:

You're forgetting to factor in time. You've already gone the first 30 km in one hour. So if you spent another hour driving at 90 km/h, you'd go 90 km. But you only need to go 30 km, which is 1/3 of 90 km, so you spend 1/3 of one hour driving the rest of the way.

Your total distance is now 60 km, and your total time is 1 1/3 hour, or 4/3 hour to simplify the math. 60/(4/3) = 45 km/h
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### #37 Incognitum

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Posted 11 July 2007 - 08:34 AM

For those saying the answer is 90 km/h:

You're forgetting to factor in time. You've already gone the first 30 km in one hour. So if you spent another hour driving at 90 km/h, you'd go 90 km. But you only need to go 30 km, which is 1/3 of 90 km, so you spend 1/3 of one hour driving the rest of the way.

Your total distance is now 60 km, and your total time is 1 1/3 hour, or 4/3 hour to simplify the math. 60/(4/3) = 45 km/h

Apparently you've not read the whole thread so I'll catch you up to speed:

Yes, we know.

We are problem solving around the question by suggesting that since there is no implicit time limit stated in the question, it is possible to bring your average speed up by driving a longer distance then is strictly necessary to reach your destination. I find the idea of a logic puzzle with the answer "it's impossible" an affront to human ingenuity.
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### #38 catdogman

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Posted 11 July 2007 - 10:54 PM

90,,, would seem the correct answer. you did half at 30 and half at 90 to find average you add them together getting 120 and divide by 2 giving 60

But lets say it is 180 miles... and you drove half at 30 taking 3 hours, the other half at 90 taking one hour... 180 miles in 4 hours or an average of 45...

So, saying you take 3 hours to do the first 90 miles, how long do you have to do the second half... 180 divided by 3 hours is 60mph... and you have already used all your time... so, not possible....

So, that is the math.. but it makes no sense.... that you cant make up the time... there is none left... zero.... if you use one minute you cant get to the 60mph average..
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### #39 unreality

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Posted 11 July 2007 - 11:20 PM

there are 2 solutions:

1) It's impossible, if you are going in the straight line that's 120 km from A to B

2) If you take a detour to even it out, you can do it. The simplest way is to extend your trip so the second half is 90 km not 60 km, and then go at 90 kmh. I think that works. Well its something like that. You can do longer detours, too, if you go faster accordingly.
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### #40 SirLogiC

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Posted 12 July 2007 - 02:06 PM

Speeding up - Back to the Logic Puzzles
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)

Spoiler for Solution

It is indeed impossible. For those that need profe I worked it out.

After working it out for a bit

even traveling at 60,000km/h for that last 30km the average speed still only reaches 59.97km/h. One thing people arent realising is that as you travel that last 30km faster you travel for less time. If you travel that last time for 60km/h you arent traveling for 1 hour but 30km. That puts your average speed to 40km/h. You do in fact have to travel at the speed of light to get the average speed.

If you really must work it out yourself the hard way (like I did :/) here is the math
choose a (speed)
now divide the 30km you need to travel by that speed
multiply this number by 60
now divide 60 by this number
write it down
multiply 30 (km/h) by this number
the number you wrote down, add one and divide your last number by this number

6000km/h
30 / 6000 = 0.005
0.005x60 = 0.3 (this number means it took you 0.3 minute to travel 30km)
60/0.3=200 **
200
200x30=6000
6000+6000=12000
12000/201=59.7km/h average speed

** I didnt do this maths in school. I dont know how to describe what Im doing here. I just know you need take into account that you have a variable (time it takes to travel 30km) that is dependant on another variable (speed you did that 30km at) and convert the initial 30km at 30km/h into equal portions based on that last 30km. Then you can just add the total portions and divide by the whole for the final average speed.

Trying to figure this out was really hard.
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