Good afternoon!
First let me make a comment about 'creating' a two digit number. For example, let's look at 85. For the 8 to be 80, we have to multiply the 8 by 10. Then we add the 5 to get 85. I am going to use that idea to create our two digit number.
Let (x y) be the two digit number for which we are searching.
Then (x y) = 10x + y.
When you reverse the digits of the number, you get
(y x) = 10y + x.
From the first sentence of the problem, we want the sum of the digits to be 5. That is,
x + y = 5 {Equation 1}
For the second sentence of the problem, we want:
(10x + y) + 27 = (10y + x)
→ 10x + y + 27 = 10y + x, by just dropping the parentheses
→ 9x - 9y = -27, by subtracting 10y and x from both sides
→ x - y = -3, by dividing through by 9
→ 2x = 2, by adding the previous equation to {Equation 1}
→ x = 1
→y = 4, since the sum of the digits equals 5.
So the two digit number for which we are looking is 14.
As a check, if you add 27 to 14, you get 41.
While you could just do trial and error, the point of this kind of question is an exercise in solving Systems of Linear Equations, and being able to translate a story problem into that system.
So, as a side note, to get a two digit number, there are only two ways to add up to 5 using digits, with the original number being smaller than when you interchange the digits,
either 1 + 4 = 5 or 2 + 3 = 5.
But 23 and 32 are only apart by 9, not 27. So we want 14 to be the original number.
Hope the above helps, and thank you for the question.
Michael Ehlers
