Jerome swapped the digits of a two-digit number, and the new number he got was 30 less than the original number. What could the original number have been?

Question

Jerome swapped the digits of a two-digit natural number, and the new number he got was 30 less than the original number. What could the original number have been?

Answer 1

Any 2 digit positive integer can be written as 10(tens digit) + units digit

 

For example, 47 = 10(4) + 7

 

Let's say that the original number is 10X + Y.  Then the number obtained after switching digits is 10Y + X.

 

According to the statement of the problem, the number obtained after switching digits is 30 less than the original number.  This fact yields the equation:  10X + Y = 10Y + X + 30.

 

So, 9X - 9Y = 30

 

     9(X - Y) = 30

 

        X - Y = 3.3333...

 

But, X and Y are integers, so X - Y must also be an integer.  This contradicts the statement that X - Y = 3.33333...

 

Conclusion:  There is no such 2 digit number. 

 

Answer 2

If you let A be the first digit and B be the second, then the original value is 10A + B and the second is 10B + A. You also know from the question that 10A + B = 10B + A + 30. From there, you can infer that 9A = 9B + 30. Since A and B are both integers in the range of 0-9, it isn't a difficult step from there to determine the values.