Question
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
Answers
GPT-4o mini
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:
- \( 254 \)
- \( 344 \)
7. **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**.
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:
- \( 254 \)
- \( 344 \)
7. **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**.