Correct option is A
Solution:
Step 1: Define the Number
Let the three-digit number be represented as xyz, where:
- x is the hundreds digit,
- y is the tens digit,
- z is the units digit.
The number in numeric form is: 100x + 10y + z.
The reverse of the number is: 100z + 10y + x.
According to the problem: (100x + 10y + z) + (100z + 10y + x) = 1089.
Step 2: Simplify the Equation
Expanding the sum: 100x + 10y + z + 100z + 10y + x = 1089.
Combining like terms: (100x + x) + (100z + z) + (10y + 10y) = 1089.
This simplifies to: 101x + 101z + 20y = 1089.
Factoring out 101 from the first two terms: 101(x + z) + 20y = 1089.
Step 3: Solve for x + z
Since 101(x + z) must be a multiple of 101, dividing 1089 by 101 gives:
1089 ÷ 101 = 10.79.
Since this is not an integer, we check integer values for x + z that make the sum a multiple of 101.
From estimation: x + z = 9.
Now, substituting x + z = 9 into the equation:
101(9) + 20y = 1089.
909 + 20y = 1089.
20y = 180.
y = 9.
Step 4: Verify the Answer
- The middle digit of the number is y.
- We found y = 9.
Thus, the correct answer is:
Final Answer: (a) 9
