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

Question

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

Show Answer

Official Answer

A

A. 4.5
B. 5
C. 5.5
D. 6
E. Data insufficient

IanStewart · Accepted Answer

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).

ArtyomYom · Answer

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

vishumangal · Answer

I guess the most easiest method to solve this Ques is to evaluate the options

nick1816 · Answer

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.

effatara · Answer

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.

sting8 · Answer

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

A_Nishith · Answer

s= d/t
Given,  s=ab/10   -------eq.1
   s= ba/12 = (ba-ab)/2   ------- eq.2
  &

Equating eq.1 & eq.2, we get
ab/10 = (ba - ab)/2
=> 10a + b = 5  
=> 55a= 44b
=> 5a = 4b  
 As 'a' and 'b' should be single digit and non-zero (as otherwise both milestones wont be double digit such that ba > ab)
=>  a=4 & b=5

So, s= (10a+b)/10 = 45/10 =  4.5 miles/hr

Answer: A

A_Nishith · Answer

s= d/t
Given,  s=ab/10   -------eq.1
   s= ba/12 = (ba-ab)/2   ------- eq.2
  &

Equating eq.1 & eq.2, we get
ab/10 = (ba - ab)/2
=> 10a + b = 5  
=> 55a= 44b
=> 5a = 4b  
 As 'a' and 'b' should be single digit and non-zero (as otherwise both milestones wont be double digit such that ba > ab)
=>  a=4 & b=5

So, s= (10a+b)/10 = 45/10 =  4.5 miles/hr

Answer: A

Shayan08 · Answer

Let for first 10 hours he travels = 10x + y miles

For next 2 hours he travels = 10y + x - 10x - y = 9y-9x miles

As speed is same:

(10x+y)/10 = (9y-9x)/2 solving eqn results:

5x=4y

Now x and y have to be single digited. To find out values of x and y, determine their LCM which is 20 and,
For 5x=4y , x=4 and y=5. Notice that if we take any other common multiple the results exceed single digits, for example: With 40, x= 8 and y=10 (not acceptable)

Proceed the equation, (10x+y)/10 = 10*4+5/10 = 4.5 miles per hour. So answer A is correct

eega · Answer

Here we can evaluate each choice one by one
taking a 4.5 mph
at first milestone distance travelled during 10 hrs will be indicated which is 4.5 * 10 = 45 miles
at second milestone after 12 hrs distance travelled 54 miles will be indicated

Hence  option A  is the answer

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards