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When the 3 digit number is rounded up to the nearest 108 miles to 3 hundred, the digit in the one place is 4. The sum of the digit in the 3 digit number is 11. What is the number

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Let's break down the problem step by step.

  1. Definition of the Problem: We are looking for a 3-digit number \( xyz \), where:

    • When rounded to the nearest 108, it rounds to 300.
    • The sum of the digits \( x + y + z = 11 \).
    • The digit in the one place \( z = 4 \).

  2. Rounding Condition:

    • A number rounds to 300 when it is between 252 and 348. This is obtained because:
      • 300 - 54 = 246 (the lower threshold),
      • 300 + 54 = 354 (the upper threshold).
    • Since we are dealing with rounding to the nearest 108: \[ 252 \leq xyz < 348 \]

  3. Digit in the Ones Place:

    • Since \( z = 4 \), we can rewrite \( xyz \) as \( 100x + 10y + 4 \).

  4. Sum of the Digits:

    • Given \( x + y + 4 = 11 \), we can deduce: \[ x + y = 11 - 4 = 7 \]

  5. Finding Suitable Digits: We need \( xyz \) within the range of 252 to 348 (inclusive) while satisfying \( x + y = 7 \) and \( z = 4 \). Thus, we can express our number as \( 100x + 10y + 4 \).

  6. Exploring Possible Values: Next, let’s find pairs of \( (x, y) \) that satisfy \( x + y = 7 \):

    • \( (0, 7) \) - Not valid (not a 3-digit number).
    • \( (1, 6) \) - Leads to \( 164 \). Check if it's within the range: \( 164 < 252 \) (not valid).
    • \( (2, 5) \) - Leads to \( 254 \). Check: \( 254 \) is in the range.
    • \( (3, 4) \) - Leads to \( 344 \). Check: \( 344 \) is in the range.
    • \( (4, 3) \) - Leads to \( 434 \) - Not valid (not a 3-digit number).
    • Higher combinations will exceed 3 digits or will not fulfill the conditions.

From the pairs that meet the rounding condition, we have:

  • \( 254 \)
  • \( 344 \)
  1. Final Validation: We need to check both valid numbers:

  • For 254:

    • The digits are \( 2, 5, 4 \).
    • Sum: \( 2 + 5 + 4 = 11 \). Condition met.
    • Range: 252 ≤ 254 < 348. Condition met.

  • For 344:

    • The digits are \( 3, 4, 4\).
    • Sum: \( 3 + 4 + 4 = 11 \). Condition met.
    • Range: 252 ≤ 344 < 348. Condition met.

Thus, both conditions are satisfied for \( 254 \) and \( 344 \). Both are valid according to the problem requirements.

The 3-digit numbers that meet all the criteria stated in the problem are 254 and 344.