A three-digit number is such that when its reverse is subtra

Question

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

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

Bunuel · Accepted Answer

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}.

Answer: C.

TGC · Answer

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

KarishmaB · Answer

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.

Answer (C)

sashibagra · Answer

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.

Bunuel · Answer

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.

Hope it's clear.

sashibagra · Answer

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 :-D

HappyQuakka · Answer

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.

gracie · Answer

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

Achuuuuuuu · Answer

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

gracie · Answer

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

exc4libur · Answer

ABC-CBA=297
100A+10B+C-100C-10B-A=297…99A-99C=297…A-C=3
3B=A-C…3B=3…B=1
A-C=3: A>1>C…3≤A≤9…9-3+1=7

Ans (C)

smmc29 · Answer

Let the number be equal to  abc

Then the reverse of the digits are  cba

Further number abc can be expressed as 100a + 10b + c
While cba can be expressed as 100c + 10b + a

We know that the difference between abc and cba is 297.

Subtract the two expressions from each other we get that 99a - 99c = 297

We can see that  a - c = 3

Given in the problem, we also have 3b = a-c; therefore,  b=9

Since the difference should be 297, we know for sure that a should be at least 3.

Let us list down all possible pairs for a and c:

{a,c} = {3,0},{4,1},{5,2},{6,3},{7,4},{8,5},and {9,6}

Hence we have a total of  7 possible numbers

And yes, we can allow c = 0. In this case the reverse of the digits will simply be a two digit integer.

bumpbot · Answer

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