The sum of the digits of a two-digit number is 13. If 27 is added to the number, the digits will be interchanged. Find the number

Question

The answer is ???

Answer 1

I'll say that the two digit number is AB where A is the tens digit and B is the ones digit (I'm not multiplying A and B).

 

AB can be rewritten as 10A + B.

 

The question tells me when I add the digits, I get 13.

 

A + B = 13

 

It also tells me that if I add 27 to the original number it reverses the digits:

 

AB + 27 = BA

 

Now by AB we actually mean 10A + B and by BA we actually mean 10B + A, so we rewrite this equation to be:

 

10A + B + 27 = 10B + A

 

Rewrite each side:

 

9A + (A + B) + 27 = 9B + (B + A)     (one way to do this is to replace any A + B s I see with 13 because of the first equation I have)

 

9A + 27 = 9B

 

A + 3 = B             (divide everything in the equation by 9)

 

Now I can substitute for B in that first equation:

 

A + B = 13

 

A + A + 3 = 13

 

A = 5

 

Then B has to add to this to give me 13, so B = 8.

 

My number is 58.

 

(Check: 5 and 8 add to 13, check. 58 + 27 = 85, check.)