Question 7795
When we have a two digit number, it can be expressed as 10a + b. Actually, let's take the number 82. That's 80 + 2 which really is 10*8 + 2. The a = 8, and the b = 2.


So, let's say that we don't know the original 2-digit number. We have to represent that as 10a + b.


The first thing they say is that when the digits are reversed, the new number is 9 more than the original number. If you reverse the digits of 10a + b, it will be 10b + a. They said that new number is 9 more. This means that the new number is greater, which then means that we must add 9 to the old number so that it will equal the new number. OLD + 9 = NEW. OLD was the 10a + b and NEW was the 10b + a. Let's plug those in: 


{{{ (10a + b) + 9 = 10b + a }}} <----- Relationship between a two digit number and the number that results in reversing the digits.


They said that the sum of the digits of the original number is 11. So a + b = 11. Let's solve for b: b = 11 - a. Now substitute 11 - a for b in the big equation:


{{{ 10a + (11 - a) + 9 = 10(11 - a) + a }}}


{{{ 9a + 20 = 110 - 9a }}} <---- Simplified


{{{ 18a = 90 }}} <---- added 9a to both sides and subtracted 20 from both sides.


{{{ a = 5 }}} <---- if a + b = 11, and a = 5, then b must be 6. If a = 5 and b = 6, 10a + b would give you 56, the original number. If you reverse 56, you get 65, which indeed is 9 more than the original number.