The Traveller
A truck travels down the hill at 72 mph, on the level at 63 mph, and up the hill at only 56 mph. The truck takes 4 hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes. What’s the distance between the two towns?
Labels: logic, mathemagic
posted by Rajesh Lal at
8:21 AM
7 Comments:
There a wide number of solutions but they all have a total of 273 miles.
a miles b miles c miles
172 97 4
173 95 5
174 93 6
175 91 7
179 83 11
180 81 12
181 79 13
182 77 14
186 69 18
187 67 19
188 65 20
189 63 21
193 55 25
194 53 26
195 51 27
196 49 28
200 41 32
201 39 33
202 37 34
203 35 35
207 27 39
208 25 40
209 23 41
210 21 42
214 13 46
215 11 47
216 9 48
217 7 49
For a more conclusive solution...
D = R * T, therefor T = D/R
if, for the trip there, we let x be downhill distance, y = level ground distance, and z = uphill distance:
x/72 + y/63 + z/56 = 4
and on the return trip:
x/56 + y/63 + z/72 = 4 + 2/3
If we multiply both equations by (7*8*9), we get:
9x + 8y + 7z = 2352
7x + 8y + 9z = 2016
add them together and get:
16x + 16y + 16z = 4368
divide by 16 and:
x + y + z = 273
Ragknot, your proof is incomplete because it only accounts for situations with integer numbers of miles. Although it does suggest that it will always add to 273, it isn't "proven"
cry wolf,
Yes, I knew there many fractional solutions, but I did not want to list an infinity of solutions, so I only listed the integers as a demonstration... but true, you showed a excellent solution.
On the first leg of the trip the distance for downhill is x, level is y and uphill is z.
Then y=441-2x and z=x-168
To get positive, non zero values for y and z, x must be between 169 and 220.
There's 168 miles more down (x) than up (y). I hope that the driver has a spacesuit.
Chris,
I see what you mean, but 168 miles uphill is not 168 miles straight up I hope.
Hi Ragknot. For the lorries to be so strongly affected, the gradient's got to be quite high. Even 1 in 20 would give about 8.4 miles altitude change.
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