Can you find SPEED of Car || Tricky GMAT exam problem
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- Опубликовано: 12 сен 2024
- This problem is from a GMAT examination.
A person travels in a car with a uniform speed.. he observes a milestone which has two digit.
After one hour, he observes another milestone with the same two digits revered.
After another hour, he observes a 3 digit milestone which has the same two digits (as that of milestone 1) and a zero between them.
The problem is :
Can you find the speed of the car?
Think mathematically and think logically.
------------------------
I have explained the solution to this GMAT math problem in both ways, a mathematical and a logical way.
I have conducted a little survey by asking this question to several persons ranging from average to excellent academics credentials. I was surprised to see that it was solved by only those people who are in between these two levels.
Those who are excellent in their academics found it very difficult to visualize and simplify the problem. However, people with distinction in their academics (in the range of 70%) solved it with a little efforts.
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I chose the second approach, but I didn't bother to solve for the mile markers, because that was redundant. M2 was halfway in between M1 and M3, and those are 1X and 10X, for some ones value X. Difference is 90, the distance in two hours; half is 45, the distance in an hour.
Same. Ignoring the second mile marker substantially simplifies the problem because the speed is the same for all values of x.
Wow, this is another brilliant way of solving the prob. Kudos.
This is how i did it too, but then I also solved for the mile markers because I forgot that all I needed was the speed.
Can you explain 1X & 10X approach?
@@destroyedsoul1791 M1=10X + Y and M3=100X + Y. Therefore, the distance between M1 and M3 is (100X + Y) - (10X + Y). That simplifies to 100X - 10X + Y - Y, which equals 90X. Since we know X=1 (per the proof in the video), then the distance = 90 miles. It takes 2 hours to travel 90 miles, so the speed is 90 / 2 = 45 mph.
This took hardly 20 seconds. Pretty easy. Used to seeing so many complicated riddles in this channel, hence this took me by surprise 😅
First digit has to be 1 as you said. And simple trick to calculate 2nd digit is to consider the fact that upon adding the same distance twice, the last digit is remaining unaltered. So definitely we are adding a number with 5 in the end (45) and the difference between the 2 digits has to be 5 as well. So 16😁
I have solved this logically exactly the way u did even before seeing the solution ,this made me so happy
Bharath.... when I tried this problem, the first thing came to my mind was the Logical approach... and it was fun to solve in mind... I hope you too solved without pen and paper :)
@@LOGICALLYYOURS me too sir same logically without paper with same thinking
Me too lol without any calculations just with logics
Same thing for me
u must be a jee aspirant? lol
45. I found it in 2 minutes. Its rare but always feels good to solve your puzzles in quick time. I chose tge logical approach 🙃🙃
(logical + mathematical approach)
after finding "1", we can minus 3rd & 1st milestone so that speed would be (100+x)-(10+x)=90 km per 2 hour so speed would be 45km/h.
This can also be asked in an interview without using mathematical approach. Very nice video!!
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.
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.
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.
I solved it logically but bit different way, when first two numbers are two digit numbers, third number must be below 200 only, as the first digit of third number is 1 and second digit is zero it will be below 110, so first number will be below 20, so the second number must be between 50 n 70 with ending 1, as 51 is not fulfill criteria 61 is the correct answer,
I could solve this interesting problem using say logic.
It is indeed a good problem for those appearing for competitive exams and interviews. If the concept is clear problem solving becomes easier.
i did same
Sometimes I used to guess the endings of movie , this time I was able to guess your logical way ....Thanks Amar ....your videos lets my mind grow 😊
Great logic. I solved this puzzle mentally within 3 minutes.
In fact you did the right job.... when we have limited calculation, we must try it in our head.
@@LOGICALLYYOURS Thank you sir
HOW MILESTONE VALUE INCREASE IF WE TRAVEL TOWARD IT.?? IT SHOULD BE IN DECREASING ORDER.
IT SHOULD BE IN SEQUENCE OF M3, M2 THEN M1?
the answer will be the same tho
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.
Got the answer before watching the solution. 😃😃
Nice puzzle :)
I used systematic approach:
x×100+y-(y×10+x)=y×10+x-(x×10+y)
Simplify
99x-9y=9y-9x
108x-18y=0
12x-2y=0
x and y should be one digit numbers so
Answer is x=1and y=6
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.
Thanks for the content
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.
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.
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.
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!!
You got it perfectly my friend!! I'm glad you liked it Tamir :)
Hi. In the logical approach, once we get to know that the first milestone is 1A and the third is 10A, gap between the two milestones is 90 for 2 hrs, giving the speed as 90/2 directly.
Thank you for the puzzle
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.
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.
1 solved it in my mind the same way you said even without looking the sol and that too in 30 sec. I was amazed by it. Very nice puzzle👍👍
Interesting problem, solved it immediately. It is indeed a nice problem useful for students appearing in various entrance exams and interviews.
Thanks Gopu sir :)
Wow , that was a quite good question👍👍👌
Wow that's awesome
Logic quickly showed X would be 1. I also saw M3 - M1 would be 90 regardless of the value of Y, which would be the distance the car traveled in 2 hours. Thus the car is traveling 45mph.
This is really very good question sir.
Very nicely explained. I also enjoy making similar puzzle videos. Thanks for sharing!
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
This is an AMAZING one.
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.
Milestones shows distance left. You travel, distance decreased.
M3, M2 and M1 makes more sense.
Same idea sir
Yes
same here...
Wow.. Logical one is awesome.....
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.
I solved this in different logic.
Since first two milestones are two digits, speed can not be three digit number. In that case all the numbers above twenty are not possible. Then it narrows down to 12-19.
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.
It was nice problem and i solved it thanks man coz of u i can observe increament in my out of box thinking
Such an amazing explanation sir 🔥
💚❤💚 This was a lot of fun. We love quizzes and riddles. You inspire us to make them and test others as well as entertaining people. Thank you.
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.
Superb !
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
Hi Ammar
I’ve got a puzzle
How can I send it to you?
Hi Rafi.... ya pls send it to me at logicreloaded@gmail.com
A very simple one. The first digit was one and the 3 nos are in arithmetic progression so the difference between 1b ( b is second digit) and 10b is 90 so spped is 90/2. Not even took 15 seconds
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.
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
I solved it without watching ur solution 👌👍
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??
Hi Ankur.... well, your doubt is absolutely valid. Typically, if you are approaching a destination, the number reduces. I found some images on Google that has numbers at the top, destination name in the middle, and then remaining distance at bottom. I have to verify if the top section number stands for progress.
Fantastic logical solution. Liked it. Keep going.
10y+x-10x-y = 100x+y-10y-x
9y-9x=99x-9y
18y=108x
y=6x
Only one possible value i.e 16
Amazing 👏👏👏👏👏
Wow I got a heart from Mr. Ammar I can't express myself
The car travels at a speed of 45 miles per hour Answer.
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.
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.
Yeah i did the same
After watching 20 times finally I understood. Thankyou
I was also also to solve it logically without watching this video . After finding answer i watched video .the method that I choose was exactly same as given in video.😍😍
Thank u sir
Interesting
I use approach 2 and using excel.
2 minutes to write the equation and less than 1 minute to get the answer.
I think approach 1 is mathematic approach and approach 2 is logic approach.
This is quite easy , solved in just half a minute.
Liked the way u showed in two diff approaches bro
Variation on logically:
min(XY)=10 ; max(YX)=99 ; max(v)=89
max(X0Y)=max(XY)+max(v)=188; and min(X0Y)=100
:: X=1 .. and max(X0Y)=109
2*min(v)=min(X0Y)-max(XY)=100-19=91
2*max(v)=max(X0Y)-min(XY)=109-10=99
min(v)=round.up(81/2)=40
max(v)=round.down(99/2)=49
min(YX)=min(XY)+min(v)=10+40=50
max(YX)=max(XY)+max(v)=19+49=68
:: Y=5 or Y=6
(Y=5):
d1=51-15=36
d2=105-51=54
(Y=6):
d1=61-16=45
d2=106-61=45=45
M1 = 10X + Y, M3 = 100X + Y, M3 - M1 = 90X, X=1. ==> 90 miles covered in 2 hours ==> speed = 45 mph.
No need to figure out Y, no need to bother with M2.
Best teacher
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
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!
Why would numbers on milestone increase if we travel in same direction ? .. they should be decreasing.
mine : M1=37, M2=73, M3=109 (distance between each milestone 36)
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
M3 uses the same two digits as M1 and M2 with a 0 in the middle. 37 is not the same as 19.
You made some assumptions that are not totally logical or correct. M2 could be smaller than M1 because mile markers start from one side and a driver could be driving from high to low. M2 had to be higher no because it would be negative speed to be a lower mile but because M3 was 3 digits. You assumed that the order of digits in M3 was the same as M1 but the problem presented never stated this. It can be shown that the order needs to be preserved in the solution but the assumption is false. You assumed that the only possible solutions were 15 and 16 because they were in the middle which was just a good place o begin looking but there was no logical reason for this assumption.
He didn't say that the car is reaching it's destination or going into a city.. This is just a study.
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.
practically, M1>M2>M3, (m1 is always greater than m2 and m2 is always greater than m3)
Very nice👍
Solved it in my mind
this is brilliant
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
I have answered it correctly through second way, before looking into answer.
I used the logical way and put a number 37 on M1,then I got 73 on M2 and finally 106 on M3. Hence I got the answer as 36.(Is this correct?)
Nice...
This was an easy question
I used algebra to solve the problem, and it worked like a charm
Debjati.... feel free to share your full approach
@@LOGICALLYYOURS sure , I assumed the first number to be A and the next to be B . So the the numerical value should be (10A+B) , thus the next value written on the second mile stone should be (10B+A) , and the third should be (100A+B).
thus the relation among these equations should be
(100A+B) - (10B+A) = (10B+A) - (10A+B)
as the distance between the first and second is same as the distance between second and third
You will get the value of A to be 1 and B to be 6
Hi sir.. but y should we take only 10x and not any other number may be 23x..Please explain
Preethi... first two milestones are only two digits each... Now think about the car speed. Car is covering a distance less than 100 miles in first hour... so it will cover less than 100 miles in next hour too.... so overall, the milestone M3 has to be less than 200... that's why it's hundreds place digit would be 1.
Even if you still have difficulty in understanding, pls let me know, I'll explain in a different way.
Because place value of number XY=10X+Y
Like 23=10(2)+3
@@LOGICALLYYOURS Thanks a lot sir.. got it....
I just said 16, 61 and 106 spontaneously without realizing that this was the solution
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.
Amazing bro
If the car is making progress then distance should decrease...but in this case distance is increasing
Am in 8th and had to do with pen & paper
In 2 minutes
I like maths 😌😁
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!?😂
But as you are traveling ahead, the numbers on milestones go down, not up
Nice
Best thing about your videos... no GOGU! :-)
😄
Question: why is it M1 = 10x+ y?
Let the 2 digit number on milestone named as M1 be XY=10X+Y
Same as 23=10(2)+3
And not M1=10x+y
000 😁. ......and car speed = 0 👍🏻
When we r travelling towards a destination the number on the milestone probably decreases ..I see🙋🙌🙎
yeah....no one is watching this basic logical fact
Hi Ammar sir
Hi Shikhar... I hope you solved and enjoyed this puzzle:)
Ammar.. I think the question should be in reverse order.. Like, first he should see 3 digit number.. As we reach destination mile stones are in decreasing order no? 🙂🙂
Think in this way. What if the car is going away from the destination which your mentioning and the mile stone tells the distance to that destination
@@alfie0 But it can't be seen by us.. we have to see back through car's window or through mirror to see that milestone 😋
@@saiprakash163 The car driver don't need to see it, knowing to the puzzle solver is enough
Algebra is much easier than your logical approach
Too bad I needed a notepad for this.