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ggarr
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 21 Mar 2007, 18:20
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55% (hard)

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66% (02:40) correct 34% (02:10) wrong based on 427 sessions

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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 level, and 1/3 downhill. If he covers the uphill part of the route at the rate of 16 miles per hour and the level part at the rate of 24 miles per hour, what rate in miles per hour would he have to travel the downhill part of the route in order to average 24 miles per hour for the entire route?

(A) 32
(B) 36
(C) 40
(D) 44
(E) 48

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Please solve and explain why this doesn't this work:
1/3(16) + 1/3(24) + 1/3(x) = 24

And don't just give me your solution. Please let me know what's wrong w/the above. This equation was the first thing that came to my mind. I need to understand what I'm missing.

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E
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KarishmaB
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve Updated on: 23 Aug 2023, 00:37
Originally posted by KarishmaB on 05 Mar 2013, 20:42.
Last edited by KarishmaB on 23 Aug 2023, 00:37, edited 1 time in total.
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ggarr
A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 level, and 1/3 downhill. If he covers the uphill part of the route at the rate of 16 miles per hour and the level part at the rate of 24 miles per hour, what rate in miles per hour would he have to travel the downhill part of the route in order to average 24 miles per hour for the entire route?

(A) 32
(B) 36
(C) 40
(D) 44
(E) 48

Please solve and explain why this doesn't this work:
1/3(16) + 1/3(24) + 1/3(x) = 24

And don't just give me your solution. Please let me know what's wrong w/the above. This equation was the first thing that came to my mind. I need to understand what I'm missing.

Please Show ALL work
Show more

Responding to a pm:

There are two problems with using the formula given on my blog on this question:

1. There are three different speeds which you need to average out.
2. The weights in case of speed is 'time taken' not distance traveled. Explained here: https://anaprep.com/arithmetic-weights- ... d-average/

In case of three speeds, you can simply use the formula: Cavg = (C1*W1 + C2*W2 + C3*W3)/(W1 + W2 + W3)

Weight in case of speed is 'time taken'

Hence, Avg Speed = Total distance / Total time (which we know)

In this question, I would like to assume that the total distance is 48*3 such that the distance traveled in each leg of the journey uphill, level and downhill is 48 miles (you can assume it to be something else or x)

Time taken to go uphill = 48/16 = 3 hrs
Time taken on level = 48/24 = 2 hrs
Time taken to go downhill = 48/d

Avg Speed = 24 = 48*3/(3 + 2 + 48/d)
48/d = 1
d = 48 miles/hr

You can also look at it in another way - A Shortcut:

Speed on level plain is 24 miles/hr and average speed is also 24 miles/hr. So we can just ignore the level plain since it is at the average.
We need to average out the rest of the journey to 24 miles/hr. Again, assuming that distance of each leg is 48 miles,

24 = 48*2/(3 + 48/d)
d = 48 miles/hr
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 21 Mar 2007, 20:51
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Average Speed= total distance/total time

24=x/((x/3*16)+(x/3*24)+(x/3*Y)
where, Y is the speed for the downhill slope.

Hence, 24=1/1/(1/16+1/24+1/3Y)

which gives, 1/24=1/16+1/72+1/3Y (not 24=1/16+1/72+1/3Y)

Solving for Y, you will get Y=48
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 22 Mar 2007, 08:50
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"Please solve and explain why this doesn't this work:
1/3(16) + 1/3(24) + 1/3(x) = 24 "

When you calculate average speed it is equal to the time traveled times the speed or if you travel half an hour at speed 60 mph and half an hour at 30 mph then the average speed would be 45 mph
If you travel 1/2h at 60mph and 1/6h at 120mph then the aver speed would be the distance traveled( 50 miles) for 40 min or av speed is 3000/40=300/4=75 mph
Hope it helps
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 22 Mar 2007, 10:35
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This is how I did it.

Take total distance = 24 miles. so each section uphill,level,downhill is 8 miles.

Now he covered uphill at 16 miles/hour so it will take him 30 minutes for 8 miles
He covered level at 24 miles/hour so he would take 20 minutes for that part

Now, overall, if he has to average 24 miles/hours for all the trip, then he has to cover 24 miles in 1 hour. So that means he has only 10 minutes to cover remaning 8 miles.
So his speed must be 8 miles in 10 minutes which is same as 48 miles/hour

As you see we dont need to write any equations and can simply do mental calculations to get to answer
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 23 Mar 2007, 18:00
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BG
"Please solve and explain why this doesn't this work:
1/3(16) + 1/3(24) + 1/3(x) = 24 "

When you calculate average speed it is equal to the time traveled times the speed or if you travel half an hour at speed 60 mph and half an hour at 30 mph then the average speed would be 45 mph
If you travel 1/2h at 60mph and 1/6h at 120mph then the aver speed would be the distance traveled( 50 miles) for 40 min or av speed is 3000/40=300/4=75 mph
Hope it helps
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You can only take the average of the three speeds if they involve the same amount of time. Clearly, they do not in this case, the leg uphill is the longest in terms of time and thus carries the greatest weight
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 18 Oct 2015, 04:54
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Let us assume that distance for each of the three parts is 48 km( Since 48 is multiple for 16 and 24 , hence it will be convenient )

Time taken to go uphill = 48/16 = 3 hours
Time taken on level = 48/24 = 2 hours
Time taken to go downhill = 48/Sd , where Sd= Speed of cyclist in downhill journey

Total time for the entire journey = Distance/ Speed
= 48x3/24
= 6 hours

Time taken to go downhill = 1 hour
Hence, Sd = 48 miles/hour
Answer E
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A cyclist rides his bicycle over a route which is 1/3 uphill, 1/3 leve 18 Jun 2021, 05:53
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