Okay, let me try to solve this problem step by step. The problem says that there's a two-digit number where the sum of its digits is 13. Then, when you subtract the reversed number from the original number (or vice versa), the difference is 27. And we need to find the sum of the original number and the reversed number. Hmm, okay, let's break this down.

First, let's denote the original two-digit number. Since it's a two-digit number, we can represent it as 10a + b, where a is the tens digit and b is the units digit. Here, a and b are integers between 0 and 9, but since it's a two-digit number, a can't be 0. So, a is between 1 and 9, and b is between 0 and 9.

The problem states that the sum of the digits is 13. So, that gives us the equation:

a + b = 13

That's straightforward. Now, the next part says that the difference between the number and the number with its digits reversed is 27. Let's think about this. The original number is 10a + b, and the reversed number is 10b + a. The difference between them is either (10a + b) - (10b + a) or (10b + a) - (10a + b). Depending on which is larger, the difference could be positive or negative. But since the problem mentions the difference is 27, which is positive, we can assume that the original number is larger than the reversed number. Therefore, the difference is (10a + b) - (10b + a) = 27.

Let me write that equation out:

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

Simplify that:

10a + b - 10b - a = 27

Combine like terms:

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

Which simplifies to:

9a - 9b = 27

We can factor out a 9:

9(a - b) = 27

Divide both sides by 9:

a - b = 3

So now we have two equations:

1. a + b = 13
2. a - b = 3

We can solve this system of equations. Let's add the two equations together:

(a + b) + (a - b) = 13 + 3

This gives:

2a = 16

Divide both sides by 2:

a = 8

Now that we have a = 8, we can substitute back into one of the equations to find b. Let's use equation 1:

8 + b = 13

Subtract 8 from both sides:

b = 5

So, the original number is 10a + b = 10*8 + 5 = 85. The reversed number is 10b + a = 10*5 + 8 = 58.

The problem asks for the sum of the original number and the reversed number. So, 85 + 58. Let me compute that:

85 + 58 = 143

Wait, let me check that addition again to be sure. 85 + 50 is 135, and then +8 is 143. Yes, that's correct.

But just to make sure we didn't make any mistakes in the process, let's verify all the conditions.

First, the sum of the digits: 8 + 5 = 13. That's correct.

Next, the difference between the original number and the reversed number: 85 - 58 = 27. That's also correct.

So, all the conditions are satisfied. Therefore, the sum of the original number and the reversed number is indeed 143.

I think that's all. Let me just recap to ensure there's no step missing. We set up the equations based on the given conditions, solved the system of equations to find the digits, then calculated the sum. All checks out. So the answer should be 143.
Therefore, the sum of the original number and the number with its digits reversed is $\boxed{143}$.
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