How many integers n greater than 10 and less than 100 a

Question

How many integers n greater than 10 and less than 100 are such that, if the digits of n are reversed, the resulting integer is n+9 ?

(A) 5 
(B) 6 
(C) 7 
(D) 8 
(E) 9

dave13 · Accepted Answer

Let A be tens digit & B be units digit X = 10A + B After reversing digits, X _{new} = 10B + A Difference between X and X_{new} is 9 , so I get following equation (10A + B) - (10B + A) = 9 9A – 9B = 9 9(A-B) = 9 A-B = 1 A = B+1 Since units digit X is +1 more than tens digit, I get following numbers 12, 23, 34, 45, 56, 67, 78, 89 IMO: D

BrentGMATPrepNow · Answer

If the digits of n are reversed, the resulting integer is n+9 ?   
Let's say n = xy (where x is the tens digit of n, and y is the units digit of n). 
So, the VALUE of n =    10x + y    (in the same way the  value  of 29 = (2)(10) + 9) 
When we REVERSE the digits of n, we get yx, which means the VALUE of yx =    10y + x

The question tells us the  resulting number  is 9 greater than the  original number (n) 
So, we can write:    10y + x   =    10x + y    + 9 
Subtract y from both sides:  9y + x = 10x + 9 
Subtract 10x from both sides:  9y - 9x = 9 
Divide both sides by 9 to get:  y - x = 1

This tells us that y, the UNITS digit of n, must be one greater than x, the TENS digit of n. 
So the possible values of n are: 12, 23, 34, 45, 56, 67, 78, and 89

Answer: D

vanam52923 · Answer

Let n=ab(as it is 2 digit number bcz >10 and less than 100)
so reverse will be =ba

now as per qsn
reverse-orgn=9--------(1)

we know ab can be written as 10*a+b(54=10*5+4)
ba=10*b+a

now substituting in 1

10b+a-10a-b=9
9b-9a=9
b-a=1
so this condition will hold true only if diff of two digits is 1

so from 11 to 20
this is true only for 12(2-1=1)

then from 21-30 true for 23(3-2=1) make sure 21 is not valid as(1-2=-1 not 1) b-a=1
so this way
each set of 10 numbers has 1 number for which b-a=1
total we have 9 such sets (11-20,21-30,---91-99)
so 8 such numbers bcz in 91-99 we cant have this condition

i.e 12,23,34,45,56,67,78,89,
so 8 such numbers

KanishkM · Answer

First of all i didn't understand the question and when i did understood it, lot of time had passed, so marked it incorrect.

Lets see, 10<n<100, digits of n are reversed, resulting number will be n+9

10x +y = 10y+x +9 
9 (x-y) = 9
x-y = 1

This means that the difference between the tens digit and units digit is 1
I messed up where ??, i had to consider these as well 
12
 23 
34
 45 
56
 67 
78
89

Now if i took 98 it wont give 108 after reversing, so out

kumarankit01 · Answer

The two digit number (assume) = 10a+b
then by the given condition
 10a+b = 10b+a+9 (digits are reversed and 9 added)
=> 9a - 9b = 9
=> a - b =1

we have following number between 10 and 100 which satisfy above condition 
12, 23 , 34, 45, 56,67,78,89

gracie · Answer

note pattern: 12/21, 23/32, 34/43...
12+11(n-1)<100
11n<99
n<9
n=8
D

GMATMBA5 · Answer

10A+B=n
10B+A=n+9

10B+A=10A+B+9
9B-9A=9
B-A=1
Integers between 11 and 99, that have unit digit 1 greater than tenths digit-12, 23, 34, 45 ..... = total 8

Ans - D
-----------------------------------------------------------------------------------------------------
 Kudos if helpful!

Mansoor50 · Answer

and range of a and b is 1 to 10?

Kinshook · Answer

Asked: How many integers n greater than 10 and less than 100 are such that, if the digits of n are reversed, the resulting integer is n+9 ?

Let the number be xy where x is tenth digit and y is unit digit.

(10y+x) - (10x+y) = 9(y-x) = 9
y-x = 1

Numbers = {12,23,34,45,56,67,78,89}

IMO D

altairahmad · Answer

Using  X = 10A + B  that you started with, gives reverse of the numbers you have mentioned. The numbers will be 21, 32, 43, 54, 65, 76, 87, 98.

To get the result that you have mentioned, your original equation should be  X_{new}  -  X  =  9

Aeyvei · Answer

0-9, since a and b are digits.

Kimberly77 · Answer

Great explanation  BrentGMATPrepNow . To clarify, is y still the unit digit after the reverse in 10y + x ? As I'm a bit confused in final values of y-x = 1 for y as unit or tens digit ? Thanks

BrentGMATPrepNow · Answer

If n = xy (where x and y are digits), then x is the tens digit of n, and y is the units digit of n.
When we reverse the digits of n, we get yx, in which case y is the tens digit, and x is the units digit.

Since the question asks us to find of the total number of values of n, we must use the information about n (it equals xy, in which case x is the tens digit, and y is the units)

Kimberly77 · Answer

Thanks BrentGMATPrepNow . Crystal clear now

barttomk · Answer

Why is the answer 8 and not 5 (12/21 , 23/32 , 34/43 , 45/54 , 56/65)? If your n is 65 and you flip it you get 56 but that is n-9 not n+9. Same goes for 76, 87, and 98

Bunuel · Answer

Please review the discussion above. The numbers are:

12, 23, 34, 45, 56, 67, 78, and 89.

rushimehta · Answer

Let's use some trial/error and logic.

What's given - when we reverse the digits of a 2 digit number, the new number - old number = 9... 
Let's now think, when is the above scenario possible - the difference, 9, is a single digit difference... so if the tens and units digits of the number have a difference of 1 between them - then only the condition given in question will be satisfied. 
If the difference between the tens and units digit is more than 1, when we reverse the digits the difference between 2 numbers is greater than 10 - simply because the tens digit goes up by 2 units.

Now let's use some trial and error... 
taking 12 --> reverse is 21 --> difference is 9. 
let's take 13 (note, the difference between the digits is 3-1=2)... --> reverse is 31 --> difference is 18. 
This confirms our above logic.

So, we can now understand that the 2 digit numbers which can satisfy the above case will have a difference of 1 between the tens and units digit, and since its given that the new number is 9 greater than the old number... we will only consider cases in which the value of number increases on reversing the digits i.e. numbers where tens digit is 1 lower than the units digit - 12, 23, 34, 45, 56,, 67, 78, 89...

How many integers n greater than 10 and less than 100 a

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