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Digits.

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by lenagmat » Mon Oct 17, 2011 2:53 am
A person travels in a car with uniform speed. He observes a mile-stone,
which has 2 digits. After one hour he observes another mile-stone with same digits reversed. After another hour he observes another mile-stone with same 2 digits containing a 0. Find the speed of the car?
(for purposes of clarity, let me just add that mile-stones tell you how far you are from your destination OR how far one has travelled from the starting point)

A. 40 mph
B. 45 mph
C. 50 mph
D. 55 mph
E. 60 mph
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by shankar.ashwin » Mon Oct 17, 2011 3:20 am
1st Milestone - 10X+Y
2nd Milestone - 10Y+X
3rd Milestone - 100X+Y

Speed/Hr = (2) - (1) = 9(Y-X)

Distance in 2 hrs = Time * Speed = 18(Y-X)

Milestone (3) = Distance travelled + Initial Milestone Reading

100X + Y = 18(Y-X) + 10X + Y

Y = 6X

Therefore, X=1,Y=6.
Corresponding Milestones would be (16,61 and 106) and speed would be 45 (61-16). B
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by GMATGuruNY » Mon Oct 17, 2011 3:50 am
lenagmat wrote:A person travels in a car with uniform speed. He observes a mile-stone,
which has 2 digits. After one hour he observes another mile-stone with same digits reversed. After another hour he observes another mile-stone with same 2 digits containing a 0. Find the speed of the car?
(for purposes of clarity, let me just add that mile-stones tell you how far you are from your destination OR how far one has travelled from the starting point)

A. 40 mph
B. 45 mph
C. 50 mph
D. 55 mph
E. 60 mph
When the tens digit and the units digit of the 1st milestone are reversed, the result must be a GREATER 2-digit integer -- 19 becoming 91, for example -- so that the 2nd milestone indicates a greater distance than the 1st milestone.
Thus, the tens digit must be LESS than the units digit.
If the tens digit is greater than 1, then the 3rd milestone will be a 3-digit integer greater than 200, yielding too great a distance between the 2nd milestone and the 3rd milestone.
Thus, the tens digit = 1.

Let U = the units digit.

1st milestone = 10 + U.
2nd milestone = 10U + 1.

3rd milestone (which contains 0):
The HUNDREDS digit must be 1; otherwise, the distance between the 2nd milestone and the 3rd milestone will be too great.
The TENS digit must be 0, implying a distance between 100 and 109; if the units digit is 0, then the distance between the 2nd milestone and the 3rd milestone will be too great.
Since the hundreds digit = 1 and the tens digit = 0, the UNITS digit = U (the units digit in the 1st milestone).
Thus, the 3rd milestone = 100 + U.

Since the car is traveling at a uniform speed, 3rd - 2nd = 2nd - 1st:
(100+U) - (10U+1) = (10U+1) - (10+U)
99 - 9U = 9U - 9
9(11-U) = 9(U-1)
11-U = U-1
12 = 2U
U=6.

Thus:
1st milestone = 16.
2nd milestone = 61.
3rd milestone = 106.
3rd - 2nd = 2nd - 1st = 45.

The correct answer is B.
Last edited by GMATGuruNY on Mon Oct 17, 2011 8:03 am, edited 3 times in total.
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by lenagmat » Mon Oct 17, 2011 4:56 am
shankar.ashwin wrote:1st Milestone - 10X+Y
2nd Milestone - 10Y+X
3rd Milestone - 100X+Y

Speed/Hr = (2) - (1) = 9(Y-X)

Distance in 2 hrs = Time * Speed = 18(Y-X)

Milestone (3) = Distance travelled + Initial Milestone Reading

100X + Y = 18(Y-X) + 10X + Y

Y = 6X

Therefore, X=1,Y=6.
Corresponding Milestones would be (16,61 and 106) and speed would be 45 (61-16). B

Why - 3rd Milestone - 100X+Y - ?
We know that it is the same 2 digits containing a 0, but zero can be at the end and then it wii be
100X+10Y+0=100X+10Y.
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by shankar.ashwin » Mon Oct 17, 2011 5:02 am
That would also be a possibility;
But use an extreme case to see if its possible. The extreme case would be 19,91.

If 0 were to be the units digit, we could have 190 in the last board.

But 91-19 will not be equal to 190-91. Hence having a 0 in the units place would give us a number which is too big (since the speed remains constant) and hence we do 100X+Y
Why - 3rd Milestone - 100X+Y - ?
We know that it is the same 2 digits containing a 0, but zero can be at the end and then it wii be
100X+10Y+0=100X+10Y.
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