Two different two-digit natural numbers are written beside each other such that the larger number is written on the left. When the absolute difference of the two numbers is subtracted from the four-digit number so formed, the number obtained is 5481. What is the sum of the two two-digit numbers?

Options

A. 70
B. 71
C. 72
D. 73

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the two-digit numbers
Let the two two-digit numbers be represented as \( x \) (the larger number) and \( y \) (the smaller number).

### Step 2: Form the four-digit number
When these two numbers are written beside each other, the four-digit number formed is \( 100x + y \).

### Step 3: Set up the equation
According to the problem, when the absolute difference of the two numbers is subtracted from this four-digit number, the result is 5481. Therefore, we can write the equation:
\[ (100x + y) - |x - y| = 5481 \]

### Step 4: Simplify the equation
Since \( x \) is larger than \( y \), we can replace \( |x - y| \) with \( x - y \):
\[ (100x + y) - (x - y) = 5481 \]
This simplifies to:
\[ 100x + y - x + y = 5481 \]
\[ 99x + 2y = 5481 \]

### Step 5: Rearranging the equation
Now, we can rearrange the equation to isolate \( y \):
\[ 2y = 5481 - 99x \]
\[ y = \frac{5481 - 99x}{2} \]

### Step 6: Determine possible values for \( x \) and \( y \)
Since \( x \) and \( y \) are two-digit numbers, we need to find values of \( x \) such that \( y \) remains a two-digit number.

### Step 7: Check for valid \( x \) values
To ensure \( y \) is a two-digit number:
1. \( 5481 - 99x \) must be even (since \( y \) is divided by 2).
2. \( 5481 - 99x \) must be greater than or equal to 20 (to ensure \( y \) is at least 10).

### Step 8: Calculate possible values
We can check values of \( x \) starting from 99 down to 10:
- For \( x = 55 \):
\[ y = \frac{5481 - 99 \times 55}{2} = \frac{5481 - 5445}{2} = \frac{36}{2} = 18 \]
Thus, \( x = 55 \) and \( y = 18 \) are valid two-digit numbers.

### Step 9: Calculate the sum
Now, we can find the sum of the two numbers:
\[ x + y = 55 + 18 = 73 \]

### Conclusion
The sum of the two two-digit numbers is \( 73 \).

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