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Senior Manager  D
Status: PhD-trained. Education, Research, Teaching, Training, Consulting and Advisory Services
Joined: 23 Apr 2019
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Difficulty:   45% (medium)

Question Stats: 73% (02:15) correct 27% (02:29) wrong based on 26 sessions

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When the digits of a two digit natural number are interchanged, then the original number is greater than four times the new number so obtained. How many such natural numbers satisfy the given conditions? Ignore numbers which have '0' in their unit's place.

(A) 2

(B) 3

(C) 5

(D) 6

(E) 7
Manager  S
Joined: 24 Sep 2015
Posts: 57
Location: Spain
Concentration: Strategy, Entrepreneurship
GPA: 3.9
WE: Management Consulting (Consulting)
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This is my first post so I'm kind of nervous Since I started studying the quant part I realized from the very beginning that people who wish to score +700 (my objective is to score 740-760) should be very good at finding more than only one way to solve a problem (specially for +700 questions). If you can quickly figure out several ways to approach the question, the chances of using the fastest way are higher. So I would recommend everyone who is targeting +700 to always train your mind to find different ways to solve problems:

1) Solve the problem in less than 2 minutes (if it's a very hard one and you think you can solve it under 3 minutes, then you can go up to 3 minutes, if not, make an educated guess and move on!)

2) If you were not able to find a second way to approach the problem while doing it, then spend some time to find it (even if you have already answered the question correctly!)

3) Solve the problem following the second approach. Write down both times and keep track of them

4) After solving +20 problems, you will be able to see what way usually works faster for you

By doing all this, you won't just realize what way works faster for you, but you will also start developing an habit, which will help you decide on more efficient ways to solve questions while being under pressure. This will increase both your accuracy and your speed in return

Now let's go ahead with this question.

I found these 2 ways in this order:

1) SOLVE IT USING SMART NUMBERS

When I foresee a problem may get a bit tricky using algebra, I always check if using smart numbers may be faster. Smart numbers are explained in MGMAT guides (i.e. guide 1). They are just numbers that are chosen by you to solve a problem, not random numbers, but numbers that you think will speed up the solving process (for instance, 100 is usually a good number for percentage problems)

After reading this problem I thought I would probably solve it faster using smart numbers.

Let's solve this!

the problem asks how many two-digit integers are higher than four times the reverse number.

Let's stop here for a second. If we chose to use smart numbers, we should quickly spot that the number that should fulfill this condition better is 91. In other words, the number with the tenth digit being the highest possible (9) and the unit digit being the lowest (1, because 0 is not allowed!)

Does 91 work? Let's check it out -> 91>4*19? --- YES because 91>76

Ok, we already got one. Now let's play with the next best numbers that can possibly fulfill this condition. We have 2 options:

1) we increase the unit digit by one -> 92
2) we reduce the tenth digit by one ->81

Which option is better. Well, it's not difficult to realize that 4 times the reverse number of the first option gives a result over 100! (29*4>100)

So we should try out the tenth unit reduction. Let's do 81, 71, 61

81>4*18? YES 81>72
71>4*17? YES 71>68
61>4*16? NO! 61<64 - Stop here: 51, 41 and so on will also have the same result!

Therefore, we have 3 numbers that fulfill the given condition. We proved that 92, 82, etc will give results over 100 so it seems there is no other possibilities (remember that unit digist cannot be 0, otherwise we would have all the numbers from 1 to 9 fulfilling the given conditions)

Option B) is correct

2) USING ALGEBRA

Let's create two variable and write the given condition in an equation

x=tenth digit of original number
y=uniti digit of original number

10x + y > 4*(10y + x)

10x + y > 40y + 4x

6x > 39y

x>6.5y

Ok, what does x>6.5y tell us? It might seem a bit difficult to infer anything from here, but it is actually not. First of all, we know that x and y cannot be higher than 9 and that both variables are integers. Therefore y can only be 0 or 1 (2 would make x go higher than 9), but we also know that the unit digit of y cannot be 0. Therefore, y can only be 1!!!

Next, if y equals 1 then how many integers between 1 and 9 are higher than 6.5? only 7,8 and 9!

Here we have our 3 numbers!!! 71, 81 and 91

Option B) is correct

I solved this problem in less time using the first approach, but you may find the Algebra approach faster!

but most importantly, I was able to train my habits properly. In case I find a similar problem that is faster to solve by using an algebra approach than a smart numbers one, I will have the proper habits to use the right one!  When the digits of a two digit natural number are interchanged, then t   [#permalink] 13 Aug 2019, 16:49
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