Let's find the three-digit number based on the given clues.
-
Multiple of 20: The last digit must be 0 since it is a three-digit number.
-
Hundreds digit and tens digit add to make 9: Let the hundreds digit be \(H\) and the tens digit be \(T\). Therefore, \(H + T = 9\).
-
Half of the number is less than 100: Since the number is three digits, the possible range is from 100 to 999. Therefore, half of our number must be less than 100, which implies the number itself must be less than 200. This restricts our three-digit number to the range of 100 to 199.
Given these conditions, we can denote the three-digit number as \(H \cdot 100 + T \cdot 10 + 0\) (since the last digit is 0).
Now, let's explore the possible values that satisfy \(H + T = 9\):
- If \(H = 1\), then \(T = 8\) → Number = 180
- If \(H = 2\), then \(T = 7\) → Number = 270 (not valid, more than 199)
- If \(H = 3\), then \(T = 6\) → Number = 360 (not valid, more than 199)
- If \(H = 4\), then \(T = 5\) → Number = 450 (not valid, more than 199)
- If \(H = 5\), then \(T = 4\) → Number = 540 (not valid, more than 199)
- If \(H = 6\), then \(T = 3\) → Number = 630 (not valid, more than 199)
- If \(H = 7\), then \(T = 2\) → Number = 720 (not valid, more than 199)
- If \(H = 8\), then \(T = 1\) → Number = 810 (not valid, more than 199)
- If \(H = 9\), then \(T = 0\) → Number = 900 (not valid, more than 199)
From the above options, the only valid candidate for a three-digit number within the number constraints is 180.
Thus, the answer is 180.