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Manager  Joined: 25 Jan 2010
Posts: 99
Location: Calicut, India
A three-digit number is such that when its reverse is subtra  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 67% (02:57) correct 33% (03:16) wrong based on 504 sessions

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A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

(A) 4
(B) 5
(C) 7
(D) 8
(E) 9

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Originally posted by cleetus on 10 Feb 2010, 06:42.
Last edited by pushpitkc on 18 Mar 2018, 10:04, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
Math Expert V
Joined: 02 Sep 2009
Posts: 58422
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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14
3
cleetus wrote:
Plz illustrate how to solve this equation question
1) A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

A) 4
B) 5
C) 7
D) 8

Hi, welcome to the Gmat Club.

Solution to your question is as follows:

Three digit number $$abc$$ can be represented as $$100a+10b+c$$, its reverse number would be $$cba$$ or $$100c+10b+a$$.

Given: $$100a+10b+c-(100c+10b+a)=297$$ --> $$a-c=3$$, this gives us 7 values for $$a$$ and $$c$$: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Also given: $$3b=a-c$$, from above we know $$a-c=3$$, hence $$3b=3$$ --> $$b=1$$, only one value for $$b$$.

So total of 7 such numbers are possible: {916}{815}{714}{613}{512}{411}{310}.

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Posts: 658
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29 GMAT 2: 680 Q50 V32 GPA: 3.7
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Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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1
abc - cba = 297

100a+10b+c - 100c - 10b - a = 297

99a-99c = 297

a-c = 3

also, 3b = a-c

b=1

Numbers abc

a1c

ac can take (9,6),(8,5),(7,4),(6,3),(5,2),(4,1),(3,0)

Hence 7
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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2
1
cleetus wrote:
A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

(A) 4
(B) 5
(C) 7
(D) 8

You can also use reasoning to solve it.

"thrice the tens digit is equal to the difference between its hundreds and units digits."

The difference between any two digits cannot be more than 9-0 = 9. So the tens digit can be 3 at most.
But if the difference between the other two digits is 9, their subtraction will give us something around 900. We need something around 300 so the tens digit must be 1 and the difference between the other two digits must be 3.

So the first such number you can have is 310. If you subtract 013 out of it, you get 297 - Correct.
Next you can have is 411. If you subtract 114 out of it, you will get 297 - Correct.
Next you can have is 512. If you subtract 215 out of it, you will get 297 - Correct.
and so on goes the pattern till you have 916.

So in all, you have 7 numbers.

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Intern  Joined: 07 Jul 2015
Posts: 9
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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Bunuel wrote:
cleetus wrote:
Plz illustrate how to solve this equation question
1) A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

A) 4
B) 5
C) 7
D) 8

Hi, welcome to the Gmat Club.

Solution to your question is as follows:

Three digit number $$abc$$ can be represented as $$100a+10b+c$$, its reverse number would be $$cba$$ or $$100c+10b+a$$.

Given: $$100a+10b+c-(100c+10b+a)=297$$ --> $$a-c=3$$, this gives us 7 values for $$a$$ and $$c$$: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Also given: $$3b=a-c$$, from above we know $$a-c=3$$, hence $$3b=3$$ --> $$b=1$$, only one value for $$b$$.

So total of 7 such numbers are possible: {916}{815}{714}{613}{512}{411}{310}.

this gives us 7 values for a and c: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Can you please write how we get 7 possible values? I understood your solution but only 7 possible probability part is unclear. Math Expert V
Joined: 02 Sep 2009
Posts: 58422
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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sashibagra wrote:
Bunuel wrote:
cleetus wrote:
Plz illustrate how to solve this equation question
1) A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

A) 4
B) 5
C) 7
D) 8

Hi, welcome to the Gmat Club.

Solution to your question is as follows:

Three digit number $$abc$$ can be represented as $$100a+10b+c$$, its reverse number would be $$cba$$ or $$100c+10b+a$$.

Given: $$100a+10b+c-(100c+10b+a)=297$$ --> $$a-c=3$$, this gives us 7 values for $$a$$ and $$c$$: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Also given: $$3b=a-c$$, from above we know $$a-c=3$$, hence $$3b=3$$ --> $$b=1$$, only one value for $$b$$.

So total of 7 such numbers are possible: {916}{815}{714}{613}{512}{411}{310}.

this gives us 7 values for a and c: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Can you please write how we get 7 possible values? I understood your solution but only 7 possible probability part is unclear. We got that a - c = 3: the positive difference between the hundreds and units digits of the number is 3. If a = 9 (max possible value of a), then c is 6, if a = 8, then c = 5, ..., if a = 3, then c = 0 (min possible value of c).

Hope it's clear.
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Posts: 9
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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Quote:
Hi, welcome to the Gmat Club.

Solution to your question is as follows:

Three digit number $$abc$$ can be represented as $$100a+10b+c$$, its reverse number would be $$cba$$ or $$100c+10b+a$$.

Given: $$100a+10b+c-(100c+10b+a)=297$$ --> $$a-c=3$$, this gives us 7 values for $$a$$ and $$c$$: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Also given: $$3b=a-c$$, from above we know $$a-c=3$$, hence $$3b=3$$ --> $$b=1$$, only one value for $$b$$.

So total of 7 such numbers are possible: {916}{815}{714}{613}{512}{411}{310}.

Quote:
this gives us 7 values for a and c: {9,6}{8,5}{7,4}{6,3}{5,2}{4,1}{3,0}.

Can you please write how we get 7 possible values? I understood your solution but only 7 possible probability part is unclear. Quote:
We got that a - c = 3: the positive difference between the hundreds and units digits of the number is 3. If a = 9 (max possible value of a), then c is 6, if a = 8, then c = 5, ..., if a = 3, then c = 0 (min possible value of c).

Hope it's clear.

I am sorry, it was really a silly question and I shouldn't asked it. It was very easy topic but I was just thinking it complicatedly so it was unclear to me. now it's clear , thanks Intern  B
Joined: 10 Aug 2017
Posts: 17
GMAT 1: 720 Q48 V41 Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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Hmm, had 6 been one of the choices, I would have fallen for it.
I basically used the same process to solve this problem -

ABC - CBA = 297, so Here I saw that A>C, A-C = 3
3B= |A-C|

Hence 1 is the only possible value for 1.

And possible values for A & C are: (9,6), (8,5), (7,4) (6,3) (5,2), (4,1) (3,0)

I was initially looking for 6, because I didn't think C could be 0 as CBA would be a two digit number..
but nowhere in the question stem say that the reverse of the three digit number ABC also has to be a three digit number. VP  P
Joined: 07 Dec 2014
Posts: 1224
A three-digit number is such that when its reverse is subtra  [#permalink]

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1
cleetus wrote:
A three-digit number is such that when its reverse is subtracted from it, the result is 297. Also, thrice the tens digit is equal to the difference between its hundreds and units digits. How many possible values are there for the number?

(A) 4
(B) 5
(C) 7
(D) 8
(E) 9

the difference between any 3-digit number xyz and it's it's reverse, zyx,
when divided by 99, gives the difference between x and z
e.g., 297/99=3
so if x=9, then z=6, and xyz=916 (we know the tens digit must be 1)
so we have:
916
815
714
613
512
411
310
7 possible values
C
Intern  B
Joined: 01 Jul 2018
Posts: 7
Re: A three-digit number is such that when its reverse is subtra  [#permalink]

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How do you say that if any 3 digit number and its reverse is divided by 99, gives the difference between hundredth and units digits? How did you arrive at that?

Posted from my mobile device
VP  P
Joined: 07 Dec 2014
Posts: 1224
A three-digit number is such that when its reverse is subtra  [#permalink]

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Achuuuuuuu wrote:
How do you say that if any 3 digit number and its reverse is divided by 99, gives the difference between hundredth and units digits? How did you arrive at that?

Posted from my mobile device

Hi Achuuuuuuu,
If you look at any two reversed 3-digit numbers,
you'll see that the difference between
them is always a multiple of 9 (in this case, 297).
When you divide this multiple by 9, you'll find
the quotient is always a multiple of 11 (in this case, 33).
When you divide that multiple by 11,
the quotient will always equal the difference between the
hundreds and units digits (in this case, 3).
Dividing the initial difference by 99 is a short cut.
I hope this helps.
gracie A three-digit number is such that when its reverse is subtra   [#permalink] 18 Sep 2018, 18:52
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