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Retired Moderator V
Joined: 21 Jun 2014
Posts: 1085
Location: India
Concentration: General Management, Technology
GMAT 1: 540 Q45 V20
GPA: 2.49
WE: Information Technology (Computer Software)
How many integers n greater than 10 and less than 100 a  [#permalink]

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1
12 00:00

Difficulty:   55% (hard)

Question Stats: 66% (02:03) correct 34% (02:21) wrong based on 287 sessions

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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

Project PS Butler : Question #36

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VP  D
Joined: 09 Mar 2016
Posts: 1221
How many integers n greater than 10 and less than 100 a  [#permalink]

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7
1
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 ##### General Discussion
Manager  S
Joined: 17 Jul 2017
Posts: 238
Re: How many integers n greater than 10 and less than 100 a  [#permalink]

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2
HKD1710 wrote:
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

Project PS Butler : Question #36

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
Director  G
Joined: 09 Mar 2018
Posts: 986
Location: India
Re: How many integers n greater than 10 and less than 100 a  [#permalink]

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HKD1710 wrote:
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

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
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Manager  S
Joined: 13 Oct 2018
Posts: 85
Location: India
GPA: 3.1
WE: Information Technology (Computer Software)
Re: How many integers n greater than 10 and less than 100 a  [#permalink]

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2
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
VP  P
Joined: 07 Dec 2014
Posts: 1233
How many integers n greater than 10 and less than 100 a  [#permalink]

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HKD1710 wrote:
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

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

Originally posted by gracie on 09 Feb 2019, 20:39.
Last edited by gracie on 10 Feb 2019, 10:39, edited 2 times in total.
Manager  G
Joined: 07 Aug 2017
Posts: 86
Location: India
GPA: 4
WE: Information Technology (Consulting)
How many integers n greater than 10 and less than 100 a  [#permalink]

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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
-----------------------------------------------------------------------------------------------------
Manager  B
Joined: 29 May 2017
Posts: 164
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: How many integers n greater than 10 and less than 100 a  [#permalink]

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dave13 wrote:
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 and range of a and b is 1 to 10? Re: How many integers n greater than 10 and less than 100 a   [#permalink] 02 Jan 2020, 02:58
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