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The distance between towns A and B is less than 100 miles. John starts

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nick1816
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The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   13 May 2019, 03:03
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The distance between towns A and B is less than 100 miles. John starts from A and after 10 hours comes across a milestone indicating the distance from A as a 2-digit number. He proceeds further at the same speed and 2 hours later comes to another milestone which shows its distance from A as a 2-digit number having the same digits as in the first milestone, but in the reverse order. What is John's speed in miles/hour?

A. 4.5
B. 5
C. 5.5
D. 6
E. Data insufficient
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IanStewart
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   13 May 2019, 07:50
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If the digits on the first sign are CD, then on the second sign they are DC. So after ten hours, he is 10C + D miles from town A, and after twelve hours he is 10D + C miles from town A. Subtracting the first distance from the second, in the two hours between the milestones, he must travel

10D + C - (10C + D) = 9D - 9C = 9(D - C)

miles, where D-C is an integer. Since he covers the distance of 9(D-C) in 2 hours, if D-C were 2 (or greater) his speed would be 9 (or greater, respectively), and in 12 hours he'd cover more than 100 miles. But we know he doesn't cover that distance in 12 hours, because when he reaches the second milestone, he still hasn't arrived at town B, and towns A and B are less than 100 miles apart. The only remaining possibility is that D-C = 1, so he covers 9(D-C) = 9*1 = 9 miles in 2 hours, and his speed is 4.5 miles per hour.

Or, if you can tell the two distances on the two milestones need to be very close in value, you can confidently backsolve from the answer choices (since it quickly becomes clear that answer E won't be correct).
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   13 May 2019, 18:18
v - speed
t - time
S - distance
A.....Mo(t = 10 hour, S = xy)......M1(t = 12 h., S = yx).....B

v*t = S

v*10=xy
v*12=yx

Divide one on another

10/12 = xy/yx
5/6 = xy/yx
so y can be 0 or 5, but next we see that y can't be 0 coz yx dwo-digit number

Hence, y = 5:
5/6 = x5/5x
Next we multiply 5 and 6 on numbers from 2 to 16 untill get nuber that suit condition above
x = 4
So
v*10=45
v = 4.5
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   13 May 2019, 21:32
I guess the most easiest method to solve this Ques is to evaluate the options
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The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   13 May 2019, 21:54
Yes distance between two mile stones {10x+y- 10y-x= 9(x-y)} must be a multiple of 9. John traveled the distance between 2 milestones in 2 hours, hence 2v is multiple of 9.

vishumangal wrote:
I guess the most easiest method to solve this Ques is to evaluate the options
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   14 May 2019, 21:45
Let the first and second milestones indicate xy and yx miles respectively.
10x+y=(5/6)(10y+x)
y=(5*x)/4
Since 'x' and 'y' are both digits, RHS of the above equation has to be a single digit whole number and the only value of 'x' which yields this result is 4. (If 'x' is 8, 'y' is equal to 10 which is not possible).
So the 1st and 2nd milestones are 45 and 54 miles respectively. John's speed=45/10=54/12=4.5 mph
Interestingly, the fact that the distance from A to B is less than 100 miles doesn't affect the result. It could have been more or less (but obviously more than 54 miles) and the result (John's speed=4.5 mph) would have been the same.
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   08 Aug 2021, 08:30
Let first milestone be xy
Therefore, 2nd milestone is yx

Thus, in 10 hours, (10x+y) distance is covered
in 12 hours (10y+x) distance is covered

since, it is given that speed is constant, we can write the equation as

\(\frac{(10x+y)}{10} = \frac{(10y+x)}{12}\)
\(\frac{(10x+y)}{5} = \frac{(10y+x)}{6}\)
\(60x+6y = 50y+5x\)
\(55x=44y\)
\(5x=4y\)

As x and y should be single digit and non-zero(as otherwise both milestones wont be double digit such that yx > xy)
x=4, y=5
Therefore, speed = 45 / 10 =4.5
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink] New post   07 Jul 2023, 19:57
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Re: The distance between towns A and B is less than 100 miles. John starts [#permalink]
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