To solve the problem step by step, we need to analyze the information given and set up equations based on the conditions provided.
### Step 1: Define the Numbers
Let the two-digit numbers \( M \) and \( N \) be represented as:
- \( M = 10a + b \) (where \( a \) is the tens digit and \( b \) is the units digit of \( M \))
- \( N = 10c + d \) (where \( c \) is the tens digit and \( d \) is the units digit of \( N \))
### Step 2: Understand the Reversed Numbers
When Jacob reverses the digits of \( M \) and \( N \), he gets:
- Reversed \( M = 10b + a \)
- Reversed \( N = 10d + c \)
### Step 3: Set Up the Equation
According to the problem, the total Jacob obtained by adding the reversed numbers is 9 less than the total he was supposed to get:
\[
(10b + a) + (10d + c) = (10a + b) + (10c + d) - 9
\]
### Step 4: Simplify the Equation
Expanding both sides gives:
\[
10b + a + 10d + c = 10a + b + 10c + d - 9
\]
Rearranging this leads to:
\[
10b + a + 10d + c - 10a - b - 10c - d = -9
\]
This simplifies to:
\[
9b + 9d - 9a - 9c = -9
\]
Dividing the entire equation by 9, we get:
\[
b + d - a - c = -1
\]
This can be rewritten as:
\[
a + c = b + d + 1
\]
### Step 5: Maximize the Sum of Digits of \( N \)
We need to maximize the sum of the digits of \( N \), which is \( c + d \). From the equation \( a + c = b + d + 1 \), we can express \( c + d \) in terms of \( a \) and \( b \):
\[
c + d = a - b + 1
\]
### Step 6: Set Constraints
Since \( M \) is greater than \( N \), we have \( a > c \). Also, \( a, b, c, d \) must be digits (0-9).
### Step 7: Test Values
To maximize \( c + d \), we can try different values for \( a \) and \( b \):
1. Let \( a = 9 \) (maximum possible value for a digit).
2. Then \( c \) must be less than \( a \). Let’s try \( c = 8 \):
- From \( a + c = b + d + 1 \), we have \( 9 + 8 = b + d + 1 \) which leads to \( b + d = 16 \).
- The maximum \( b \) can be is 9, so if \( b = 9 \), then \( d = 7 \) (not possible since \( b \) must be less than \( d \)).
- Let’s try \( b = 7 \), then \( d = 9 \) (not possible since \( d \) must be less than \( c \)).
- Let’s try \( b = 6 \), then \( d = 10 \) (not possible).
Continuing this process, we find suitable values for \( a \) and \( b \) that satisfy all conditions.
### Final Values
After testing various combinations, we find:
- \( a = 8 \), \( b = 7 \), \( c = 7 \), \( d = 8 \) gives \( c + d = 15 \).
### Conclusion
The maximum value of the sum of the digits of \( N \) is:
\[
\text{Maximum value of } (c + d) = 16
\]
