Can you find SPEED of Car Tricky GMAT exam problem
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Since if you let a= the ten-digit and 'b' = the unit-digit, then the first milestone TWO-DIGIT = 10a + b
the second milestone TWO-DIGIT then is 10b+ a (since the digit is reversed) and the third milestone THREE-DIGIT is 100a + b (since there is a '0 in the middle and the first and last digit is similar to the first milestone.
The difference in distance between the second milestone and the first milestone = the difference between
the last milestone and the second milestone given that it took the car 1 hour from milestone-one to milestone-two and ALSO one hour from milestone-two to milestone-three, all at a UNIFORM SPEED. Therefore
10b+a -( 10a+b)=100a + b -(10b+a)
10b-b-10+a = 100a-a-10b+b
9b-9a = 99a - 9b
9b+9b = 99a + 9a
18b = 108 a
b= 6 a ( divide both sides by 18. note if b had= 3a, then there would be more than one possible
outcome, such a 1 and 3, 2 and 6 and 3 and 9 since 3 is three times 1 and 6 is three times 2 and 9 is three
times 9, so good that it is b=6a, since just one outcome 1 and 6.
Therefore one digit is 6 times the other digit, so from these digits 1, 2, 3, 4, 5, 6 7, 8, 9. only 1 and 6 satisfies b= 6a. the lowest digit couldn't have been '2' since 2x 6= 12. so it is 1.
So the first milestone = 16 miles, the second 61miles and the last 106 miles
The distance between each milestone is 45 miles (61-16=45 and 106-61=45. Since speed equals distance/time, then the speed of the car is 45miles/1hour =
45 miles per hour Answer. This was a nice one.
Date: 2023-11-15
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Comments and reviews: 29
Athinira
Solved in approx 30 seconds in my head using logic and simple math.
First: The first digit of M1 must be lower than 2nd digit of M1. If M1 was, say, 52, then M2 is gonna be 25, which is going backwards. So digit 2 must be larger than digit 1.
Second: If the first milestone is too high, then the car will travel at speeds that are unreasonable for a car. If M1 was, say, 57, then M3 would be 507, meaning that car would be traveling at over -220 Mph, which is theoretically possible, but unlikely. So if you assume that it's a normal car that doesn't travel high speed, that once again points you in the right direction of M1 being a lower-end number.
Then it's just a matter of trying out some numbers in your head. If M1 = 18 miles, then M2 = 81 miles and M3 = 108 miles. Doesn't add up with uniform speed.
16 however. M1 = 16 means M2 = 61 and M3 = 106.
61 - 16 = 45
106 - 61 = 45
Therefore M1 = 16, M2 = 61 and M3 = 106, and the car is traveling at a speed of 45 Mph.
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Solved in approx 30 seconds in my head using logic and simple math.
First: The first digit of M1 must be lower than 2nd digit of M1. If M1 was, say, 52, then M2 is gonna be 25, which is going backwards. So digit 2 must be larger than digit 1.
Second: If the first milestone is too high, then the car will travel at speeds that are unreasonable for a car. If M1 was, say, 57, then M3 would be 507, meaning that car would be traveling at over -220 Mph, which is theoretically possible, but unlikely. So if you assume that it's a normal car that doesn't travel high speed, that once again points you in the right direction of M1 being a lower-end number.
Then it's just a matter of trying out some numbers in your head. If M1 = 18 miles, then M2 = 81 miles and M3 = 108 miles. Doesn't add up with uniform speed.
16 however. M1 = 16 means M2 = 61 and M3 = 106.
61 - 16 = 45
106 - 61 = 45
Therefore M1 = 16, M2 = 61 and M3 = 106, and the car is traveling at a speed of 45 Mph.
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Zahid
45
As the first two numbers are two digit numbers, the difference has to be less than 100, and the sum of those two digits has to be divisible by 9 (10x + y - 10y + x = 9x - 9y or 9 (x-y.
The maximum possible difference is 99 (not actually possible, but let's not argue about it.
If we add this number to the biggest two digit number, i. e, 99, the result is 188. So the maximum possible third number has the first digit of 1. So the first digit on the first milestone is 1.
The number on the second milestone is halfway between the first and the third number.
As we know that the second digit on the third milestone is 0, so the difference between the first and third is 90. So the difference between the first and second milestone is 45, which is the distance travelled in one hour. Though not asked but I can tell you that the first milestone is 16, second is 61, and third is 106.
In fact, they should be in reverse order if milestones have numbers in decreasing order.
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45
As the first two numbers are two digit numbers, the difference has to be less than 100, and the sum of those two digits has to be divisible by 9 (10x + y - 10y + x = 9x - 9y or 9 (x-y.
The maximum possible difference is 99 (not actually possible, but let's not argue about it.
If we add this number to the biggest two digit number, i. e, 99, the result is 188. So the maximum possible third number has the first digit of 1. So the first digit on the first milestone is 1.
The number on the second milestone is halfway between the first and the third number.
As we know that the second digit on the third milestone is 0, so the difference between the first and third is 90. So the difference between the first and second milestone is 45, which is the distance travelled in one hour. Though not asked but I can tell you that the first milestone is 16, second is 61, and third is 106.
In fact, they should be in reverse order if milestones have numbers in decreasing order.
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Canis_Lupus
For two 2-digit numbers to be the reverse of each other, their difference must be a multiple of 9 since since both will divide by 9 with the same remainder. So the speed of the car (or distance M2-M1) = 9-x.
And for the fist and last milestone to end in the same digit, their distance must be a multiple of 10. Due to uniform speed (M3-M1)/2 = 10y/2 = 5-y, the speed must be also at least a multiple of 5. From this we can deduce M2-M1 = 5-9-n = 45-n, where n is an integer. 0 is invalid, as this would mean no movement at all and 3 could not have 3 digits, n>=2 would make M2 larger than a 2-digit number. So the only valid solution can be n=1 or the speed of the car is 45mph.
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For two 2-digit numbers to be the reverse of each other, their difference must be a multiple of 9 since since both will divide by 9 with the same remainder. So the speed of the car (or distance M2-M1) = 9-x.
And for the fist and last milestone to end in the same digit, their distance must be a multiple of 10. Due to uniform speed (M3-M1)/2 = 10y/2 = 5-y, the speed must be also at least a multiple of 5. From this we can deduce M2-M1 = 5-9-n = 45-n, where n is an integer. 0 is invalid, as this would mean no movement at all and 3 could not have 3 digits, n>=2 would make M2 larger than a 2-digit number. So the only valid solution can be n=1 or the speed of the car is 45mph.
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Ankur
Thanks Ammar for posting this puzzle, I could solve it mathematically but logical approach was something that I learnt after watching the solution, thank you for that.
However, by the way of logical thinking I have found a logical problem with the puzzle itself, as one progresses on the road, the value of every milestone reduces, but in this puzzle, it is stated otherwise.
Shouldn't this have started with X0Y, then YX and then XY?
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Thanks Ammar for posting this puzzle, I could solve it mathematically but logical approach was something that I learnt after watching the solution, thank you for that.
However, by the way of logical thinking I have found a logical problem with the puzzle itself, as one progresses on the road, the value of every milestone reduces, but in this puzzle, it is stated otherwise.
Shouldn't this have started with X0Y, then YX and then XY?
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Ambrish
To find the speed we really need to know the distance between 1st and 3rd mile stone
1st mile stone = 10x + y
2nd mile stone = 10y + x
3rd mile stone = 100x + y
3rd -1st = 90x
speed = 45x (90x / 2) or distance between 2 mile stone = 45x
Since we know that 1st 2 mile stones have double digits which makes x = 1 and hence speed = 45.
I don't even have to figure out the values on mile stones to calculate the speed.
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To find the speed we really need to know the distance between 1st and 3rd mile stone
1st mile stone = 10x + y
2nd mile stone = 10y + x
3rd mile stone = 100x + y
3rd -1st = 90x
speed = 45x (90x / 2) or distance between 2 mile stone = 45x
Since we know that 1st 2 mile stones have double digits which makes x = 1 and hence speed = 45.
I don't even have to figure out the values on mile stones to calculate the speed.
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----
Haven't watched the video yet:
Mind Calculation:
As speed is constant, the middle marker would be equidistant from other two. And the middle marker is a two digit number, so the last marker must be 10_.
Now 1st marker must be 1_.
Now with trial & error second digit must be 5 or 6.
I tried 6 first, & it was right.
16, 61 &106.
So distance between each marker is 45 km.
Thus, the speed is 45 Km/Hr.
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Haven't watched the video yet:
Mind Calculation:
As speed is constant, the middle marker would be equidistant from other two. And the middle marker is a two digit number, so the last marker must be 10_.
Now 1st marker must be 1_.
Now with trial & error second digit must be 5 or 6.
I tried 6 first, & it was right.
16, 61 &106.
So distance between each marker is 45 km.
Thus, the speed is 45 Km/Hr.
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Ralph
Similar to your logical approach, but after determining that the first digit was 1, I stuck with the -maximum- and -minimum-- approach and looked at the fact that the M3 value could range from 102 to 109 with corresponding 12 to 19 for M1. this meant the M2 values ranged from 57 to 64. We know M2 has to end in a 1, and there is only one such value in the range 57 to 64. Thus, it is 61, making M1 16 and M3 106.
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Similar to your logical approach, but after determining that the first digit was 1, I stuck with the -maximum- and -minimum-- approach and looked at the fact that the M3 value could range from 102 to 109 with corresponding 12 to 19 for M1. this meant the M2 values ranged from 57 to 64. We know M2 has to end in a 1, and there is only one such value in the range 57 to 64. Thus, it is 61, making M1 16 and M3 106.
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education
I used different approach. The only number that can reverse the last digit back to its original number is 5: -(1, 6, 1, (2, 7, 2,, (9, 4, 9)-. I tried the pairing of 16 and 61, add the difference (45) to 61 and found 106. The other pairs are simply impossible because with the addition of 10 km/hr for each level of the pairing the first digit of the third mile would not be achievable.
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I used different approach. The only number that can reverse the last digit back to its original number is 5: -(1, 6, 1, (2, 7, 2,, (9, 4, 9)-. I tried the pairing of 16 and 61, add the difference (45) to 61 and found 106. The other pairs are simply impossible because with the addition of 10 km/hr for each level of the pairing the first digit of the third mile would not be achievable.
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David
The information about the second milestone is mostly irrelevant. The only part that is important about it is that it is a two-digit milestone. Then you know that the first milestone needs to have a 1 in the 10s digit with the repeated digits. This immediately gives you the speed of 45 miles per hour. It absolutely doesn't matter what the ones digit is for the question asked.
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The information about the second milestone is mostly irrelevant. The only part that is important about it is that it is a two-digit milestone. Then you know that the first milestone needs to have a 1 in the 10s digit with the repeated digits. This immediately gives you the speed of 45 miles per hour. It absolutely doesn't matter what the ones digit is for the question asked.
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Audarya
We are not asked to calculate the values of M1, M2, M3 but only the speed of the car.
M2-M1 = M3-M2 = 0. 5-(M3-M1) -same time & same speed through out.
Because M2-M1 < 100 (both 2-digit numbers, M3-M2 < 100 -1st digit of M1 & M3 =1.
Then M3-M1 = 90 (the unknown digit in 1's place is the same in M1, M3 which cancel out)
Speed of car = 90miles/2h = 45 mph
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We are not asked to calculate the values of M1, M2, M3 but only the speed of the car.
M2-M1 = M3-M2 = 0. 5-(M3-M1) -same time & same speed through out.
Because M2-M1 < 100 (both 2-digit numbers, M3-M2 < 100 -1st digit of M1 & M3 =1.
Then M3-M1 = 90 (the unknown digit in 1's place is the same in M1, M3 which cancel out)
Speed of car = 90miles/2h = 45 mph
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jonniboye
I started logically by figuring out the 1 right away, then switched to a mathematical solution for the rest. If M1 = 1X and M3 = 10X, then the difference between the two is 10X - 1X. X doesn't matter at that point since it's in the ones place on both sides, so you end up getting 100-10 = 90 for the distance between M1 and M3. Divide by 2 hours and you get 45mph.
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I started logically by figuring out the 1 right away, then switched to a mathematical solution for the rest. If M1 = 1X and M3 = 10X, then the difference between the two is 10X - 1X. X doesn't matter at that point since it's in the ones place on both sides, so you end up getting 100-10 = 90 for the distance between M1 and M3. Divide by 2 hours and you get 45mph.
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T4P
1st: A 3 digit sum of two 2 digit numbers always starts with -1-. Hence, M1 must be 1? and M2 =? 1. 2nd: The difference M3 - M1 must end with 0, otherwise M3 can not end with the same digit like M1. Thus, the distance = speed unit must be? 5 ( x2 =? 0. This works for only for one digit pair: 1 and 6! M1 = 16, M2= 61 and distances (mi) = speed (mi/h) = 45.
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1st: A 3 digit sum of two 2 digit numbers always starts with -1-. Hence, M1 must be 1? and M2 =? 1. 2nd: The difference M3 - M1 must end with 0, otherwise M3 can not end with the same digit like M1. Thus, the distance = speed unit must be? 5 ( x2 =? 0. This works for only for one digit pair: 1 and 6! M1 = 16, M2= 61 and distances (mi) = speed (mi/h) = 45.
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Robert
what about 37 - 73 - 109 it also matches the pattern and the gap is the same and i got the speed of 36m/hr. Is this solution also possible or am I missing something? EDIT: Yes I am in mistake i missed one of the rules I didnt see that the first figure of the first milestone has to match the first figure of the last milestone. Still a very fun puzzle!
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what about 37 - 73 - 109 it also matches the pattern and the gap is the same and i got the speed of 36m/hr. Is this solution also possible or am I missing something? EDIT: Yes I am in mistake i missed one of the rules I didnt see that the first figure of the first milestone has to match the first figure of the last milestone. Still a very fun puzzle!
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Tamir
1st M
10a+b
2nd M
10b+a
3rd M
100a+b
So we have:
(100a+b) - (10b+a) =
(10b+a) - (10a+b)
At the end we get
6a = b
Which can be true only for a=1 and b=6
Solution:
M1. 16
M2. 61
M3. 106
The car drive 90 miles in 2 hours so its speed is 45mph
ONE MORE GREAT RIDDLE FROM AMMAR!
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1st M
10a+b
2nd M
10b+a
3rd M
100a+b
So we have:
(100a+b) - (10b+a) =
(10b+a) - (10a+b)
At the end we get
6a = b
Which can be true only for a=1 and b=6
Solution:
M1. 16
M2. 61
M3. 106
The car drive 90 miles in 2 hours so its speed is 45mph
ONE MORE GREAT RIDDLE FROM AMMAR!
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Vivek
You need not to go one by one from 10 to 19,
You must understand that unit digit of first and third milestone is same, and that can only happen when difference between first and second digit of first milestone is either 0 or 5
If this doesn-t make sense, you can reply to this comment, will try to make it simpler.
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You need not to go one by one from 10 to 19,
You must understand that unit digit of first and third milestone is same, and that can only happen when difference between first and second digit of first milestone is either 0 or 5
If this doesn-t make sense, you can reply to this comment, will try to make it simpler.
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Freid
I simplified the first step in method 2.
The largest 2 digit number is 99, and the smallest non negative value that exists is 0.
99-0 is 99.
Doubling 99 (adding the two equal distances) is less than 200, therefore the first digit of the 3 digit number is smaller than 2, making X=1 the only possibility.
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I simplified the first step in method 2.
The largest 2 digit number is 99, and the smallest non negative value that exists is 0.
99-0 is 99.
Doubling 99 (adding the two equal distances) is less than 200, therefore the first digit of the 3 digit number is smaller than 2, making X=1 the only possibility.
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famberger5
I solved it logically in a different and unique way: if i add the speed twice and i get the same last digit, then the last digit of my speed has to be 5. Then i figured out that the first digit on M1 has to be 1 and if I add 5 + 1 =6. Thus i know, that my 1st digit on m2 and my last digit on m1 have to be 6.
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I solved it logically in a different and unique way: if i add the speed twice and i get the same last digit, then the last digit of my speed has to be 5. Then i figured out that the first digit on M1 has to be 1 and if I add 5 + 1 =6. Thus i know, that my 1st digit on m2 and my last digit on m1 have to be 6.
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sanket772
I was able to solve it with logical approach however the question kind of bothered me as if you are reading a milestone, it is approaching to a destination in which case numbers should decrease and not increase. It would make more sense if the M1 was the 3 digit number and M2 and M3 were two digit numbers
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I was able to solve it with logical approach however the question kind of bothered me as if you are reading a milestone, it is approaching to a destination in which case numbers should decrease and not increase. It would make more sense if the M1 was the 3 digit number and M2 and M3 were two digit numbers
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Choi
Logical thinking.
1st:
m1 &m2 max number is 99, therefore m3 max is 198.
2nd:
m3 middle digit is 0, means it is between 100 to 109.
3rd:
if m3 = 100, then m1 =10,
If m3 = 109, then m1 =19,
Means different between m1 &m3 is 90km
Calculation.
90km/2hours = 45 speed
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Logical thinking.
1st:
m1 &m2 max number is 99, therefore m3 max is 198.
2nd:
m3 middle digit is 0, means it is between 100 to 109.
3rd:
if m3 = 100, then m1 =10,
If m3 = 109, then m1 =19,
Means different between m1 &m3 is 90km
Calculation.
90km/2hours = 45 speed
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David
I did it in my head, No electronic calculator. About 5 minutes. 1st digit of M1 has to be a 1 so start with 15 cuz it is midway and adjust from there. 15 + 51 has a speed of 36 but that is too slow to get to triple digits in the next hour. When you get to 16 u r done! 61 - 16 = 45. 61 + 45 = 106.
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I did it in my head, No electronic calculator. About 5 minutes. 1st digit of M1 has to be a 1 so start with 15 cuz it is midway and adjust from there. 15 + 51 has a speed of 36 but that is too slow to get to triple digits in the next hour. When you get to 16 u r done! 61 - 16 = 45. 61 + 45 = 106.
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David
Recognizing that the last digit returns back to its original value on the last marker means that you needed to add a multiple of 10 from the first to the last marker. Divide that in 2, to get that the second digit must be 5 higher than the first. Much quicker way of getting the 6.
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Recognizing that the last digit returns back to its original value on the last marker means that you needed to add a multiple of 10 from the first to the last marker. Divide that in 2, to get that the second digit must be 5 higher than the first. Much quicker way of getting the 6.
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Eminem
Problem's basic logic is wrong sir. Basically miles numbers gets decreasing order while heading to destination. don't u think. btw lyk both the methods. d way of explaining. u hav a gd voice sir. keep up. bt kindly rechk all d facts den post a problem. best luck 4 ahead videos
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Problem's basic logic is wrong sir. Basically miles numbers gets decreasing order while heading to destination. don't u think. btw lyk both the methods. d way of explaining. u hav a gd voice sir. keep up. bt kindly rechk all d facts den post a problem. best luck 4 ahead videos
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sourround2002
I solved it not by finding the milestone values, but by taking the difference between any m1 and m3 values (90) and dividing by two. (102-12=90, 103-13=90, 104-14=90, etc. Since the riddle doesn't ask for milestone values, only the car's speed, this works as an answer.
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I solved it not by finding the milestone values, but by taking the difference between any m1 and m3 values (90) and dividing by two. (102-12=90, 103-13=90, 104-14=90, etc. Since the riddle doesn't ask for milestone values, only the car's speed, this works as an answer.
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P. B.
Did no one wonder how does the milestone increase in value as one drives? The person is driving in the wrong lane else milestones always reduce in value. This should be the first logical thought not mathematics. This would be the difference between common sense and book worm.
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Did no one wonder how does the milestone increase in value as one drives? The person is driving in the wrong lane else milestones always reduce in value. This should be the first logical thought not mathematics. This would be the difference between common sense and book worm.
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Mathematics
Solved differently, because 2 milestones are 2 digits, the 1st and 3rd numbers are 1X and 10X. Difference is 90, as speed is equal throughout, speed is 45.
Next step to find value of X, : 1X and X1 difference is 45, so X can only be 6, hence 16, 61, 106
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Solved differently, because 2 milestones are 2 digits, the 1st and 3rd numbers are 1X and 10X. Difference is 90, as speed is equal throughout, speed is 45.
Next step to find value of X, : 1X and X1 difference is 45, so X can only be 6, hence 16, 61, 106
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Amitabh
One more clue which i used to solve that addition of digit of car speed had to be 9. As digit of M1 become reversed after one hour of travelling in M2 so so addition of both digit of difference of M2 and M1 has to be 9, which is actually speed of the car.
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One more clue which i used to solve that addition of digit of car speed had to be 9. As digit of M1 become reversed after one hour of travelling in M2 so so addition of both digit of difference of M2 and M1 has to be 9, which is actually speed of the car.
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DESTROYED
With the eager to solve the puzzle, everyone failed to notice a major flaw in the question itself. if the car makes progress the distance to the destination should decrease. which means M1 should have 106miles & M3 should have 16miles written on them.
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With the eager to solve the puzzle, everyone failed to notice a major flaw in the question itself. if the car makes progress the distance to the destination should decrease. which means M1 should have 106miles & M3 should have 16miles written on them.
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Krishnan
I solved by the mathematical approach. Then I wondered why the motorist didn-t travel 45 miles in the first hour, if the motorist drove at constant speed. So a statement that that in the first one hour he didn-t travel at uniform speed? -
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I solved by the mathematical approach. Then I wondered why the motorist didn-t travel 45 miles in the first hour, if the motorist drove at constant speed. So a statement that that in the first one hour he didn-t travel at uniform speed? -
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Anshu
Why the value of M1 needs to be in between 10 and 19 what can't it be greater than 19. For example the value for M1 can be 37 and M2: 73 and M3: 109. These are valid values and satisfy all the given constraints for the question
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Why the value of M1 needs to be in between 10 and 19 what can't it be greater than 19. For example the value for M1 can be 37 and M2: 73 and M3: 109. These are valid values and satisfy all the given constraints for the question
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