A number consists of three digits of which the middle one is zero and their sum is 4. If the number formed by interchanging the first number itself…

Correct Option (2)

Let the three-digit number be represented as 100x + 10y + z, where x is the hundreds digit, y is the tens digit (middle digit), and z is the units digit (last digit).

• According to the problem statement, the middle digit is zero, so y = 0.
• The sum of the digits is 4: x + y + z = 4. Substituting y = 0, this equation simplifies to x + z = 4.
• The original number can therefore be expressed as 100x + z.
• When the first and last digits are interchanged, the new number becomes 100z + 10y + x. With y = 0, the new number is 100z + x.
• The problem states that the number formed by interchanging the first and last digits exceeds the original number by 198. This can be formulated as: (100z + x) - (100x + z) = 198
• Simplifying the equation: 99z - 99x = 198 99(z - x) = 198 z - x = 198 / 99 z - x = 2

Thus, the difference between the first and last digits is 2.

Incorrect Options:

Options 1 (1), 3 (3), and 4 (4) are incorrect. The mathematical derivation, based on the problem's conditions, establishes that the difference between the first and last digits must be 2. Any other value for (z - x) would not satisfy the equation 99(z - x) = 198, which is a direct consequence of the problem's premise.