How many 3-digit numbers satisfy the property that the digit in the middle is the average of the first and the third digit?


Answer:

45

Step by Step Explanation:
  1. Average of any two numbers a and b is equal to  
    a + b
    2
     . This signifies that, for their average to be a natural number, the sum of a and b has to be an even number, which further means that both a and b have to be either even or odd.
  2. The number of such pairs will be:

    Let the first box represents the hundreds digit and the second box represents the units digit.
    Let us first consider the case when both of them are odd. The first and the second box both have five possibilities: 1, 3, 5, 7, 9
    This means that there will be 5 × 5 = 25 such pairs.
  3. Now let us consider the case when both of them are even. The possible number of digits at the hundreds place are 2, 4, 6, 8 i.e. 4 digits (the hundreds digit of a three digit number cannot be 0).
    The possible digits for units place will be equal to 0, 2, 4, 6, 8, i.e. 5 digits.
    This means that there will be 5 × 4 = 20 such pairs.
  4. Therefore, the total number of such possible numbers is equal to 25 + 20 = 45.

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