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{:("Column A","X is a 3-digit number and...

`{:("Column A","X is a 3-digit number and Y is a 4-digit number. All the digits of X are greater than 4, and all the digit of Y are less than 5" ,"Column B"),("The sum of the digits of X" , ,"The sum of the digits of Y"):}`

A

If column A is larger

B

If column B is larger

C

If the columns are equal

D

If there is not enough information to decide

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two columns based on the given conditions for the numbers X and Y. ### Step-by-Step Solution: 1. **Understanding the digits of X and Y**: - X is a 3-digit number where all digits are greater than 4. Therefore, the possible digits for X are 5, 6, 7, 8, and 9. - Y is a 4-digit number where all digits are less than 5. Therefore, the possible digits for Y are 0, 1, 2, 3, and 4. 2. **Calculating the sum of digits for X**: - The smallest possible value for X is 555 (where each digit is 5). The sum of the digits in this case is: \[ 5 + 5 + 5 = 15 \] - The largest possible value for X is 999 (where each digit is 9). The sum of the digits in this case is: \[ 9 + 9 + 9 = 27 \] - Therefore, the sum of the digits of X can range from 15 to 27. 3. **Calculating the sum of digits for Y**: - The smallest possible value for Y is 0000 (where each digit is 0). The sum of the digits in this case is: \[ 0 + 0 + 0 + 0 = 0 \] - The largest possible value for Y is 4444 (where each digit is 4). The sum of the digits in this case is: \[ 4 + 4 + 4 + 4 = 16 \] - Therefore, the sum of the digits of Y can range from 0 to 16. 4. **Comparing the sums of X and Y**: - The sum of the digits of X ranges from 15 to 27. - The sum of the digits of Y ranges from 0 to 16. - Since the minimum sum of X (15) is greater than the maximum sum of Y (16), we can conclude that the sum of the digits of X will always be greater than the sum of the digits of Y. ### Conclusion: - Based on the analysis, the sum of the digits of X (Column A) is always greater than the sum of the digits of Y (Column B). Therefore, the answer is that Column A is larger. ### Final Answer: - **Option 1**: Column A is larger.
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