Source: Official Guide for the GMAT 13th Ed. Problem Solving; #212 Official Guide for the GMAT 2015 14th Ed. Problem Solving; #212

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# If a two-digit positive integer has its

If a two-digit positive integer has its digits reversed, the resulting integer differs from the original by 27. By how much do the two digits differ?

### 4 Explanations

1

Ashish Raj

Is there a methodology you have to answer these type of questions? The video seems more like guessing than strategy.

Dec 12, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Hi Ashish,

Excellent question! As I mention in another comment, the equation Vincent came up with is general and could be applied to situations in which we were given a different difference between ab and ba. However, it's not guaranteed that a 2-digit number that satisfies the conditions of the problem exists for all possible differences between ab and ba. When using that type of equation, it's key to remember that the digits must be integers :)

Overall, here's how we come up with those equations:

If we say that

A = the tens digit
B = the units digit.

then,

original integer = 10A + B

and

original integer with digits reversed = 10B + A

Now, we can use the fact that the difference between these two two-digit numbers is 27 to write the following equation:

10A + B - (10B + A) = 27

We have one equation and two variables. However, this is enough to answer the question, which asks for the difference between the two variables, A - B!

10A + B - (10B + A) = 27
10A + B - 10B - A = 27
9A - 9B = 27
9(A - B) = 27
A - B = 3

Again, this type of equation can be used for similar situations :)

Hope this helps!

1

Nichole Bestman

Hi! I've also learned how to do this type of problem the way Vincent did in the above explanation. But because this method was faster, I would like to know if this way is always gauranteed? For ex, could it have been 38 (two away from 40) and we get the right answer?

Oct 21, 2015 • Comment

Cydney Seigerman, Magoosh Tutor

Hi, Nichole! For this question, we're told that the difference between ab and ba is 27, and due to the restrictions in the question, we can determine that the difference between the 2 digits is 3. The equation Vincent came up with is general and could be applied to situations in which we were given a different difference between ab and ba. However, it's not guaranteed that a 2-digit number that satisfies the conditions of the problem exists for all possible differences between ab and ba. For example, if we solve for a-b when ab - ba = 38, we'll find that we do not get an integer answer, indicating that a number ab for which ab - ba = 38 does not exist.
I hope this helps!

5

Vincent Weng

It doesn't make any sense that "a.b - b.a = 27." the correct equation should be (10a + b) - ( 10b +a ) =27. Also, if a=5 b=3 the ab -ba wouldn't be 27.

(10a + b) - ( 10b + a) = 27

10a +b - 10b -a = 27

9a-9b = 27 divide by 9

a-b=3 ans is 3

Feb 28, 2014 • Comment

Lucas Fink, Magoosh Tutor

Hi, Vincent! Your method is perfectly valid—nice work creating that equation. :-)

But it seems like there's some confusion about what's said in that video. "ab" wasn't supposed to represent a*b, although I see how it might look that way. Instead, "ab" represents a number with digits a and b; they're place-holders, not a mathematical equation. The point is that since we know the difference is 27, and 27 is close to 30, there's a good chance that the difference between a and b is 3 (look only at the ten's place to see this). With a bit of number sense, you can get to the answer without creating the equation.

And in that video, Mike uses a=5 and b=2, not b=3. If a=5 and b=2, then we have 52 - 25 = 27, which does work. :-)

I hope that makes more sense, now! Mike McGarry, Magoosh Tutor