`{:("Column A","a and b are the digits of a two-digit number ab, and b = a + 3" ,"Column B"),("The positive two-digit number ab", ,"The positive two-digit number ba"):}`

Options

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

Answer

To solve the problem, we need to determine the relationship between two-digit numbers formed by the digits \( a \) and \( b \), where \( b = a + 3 \).

**Step 1: Define the two-digit numbers.**
The two-digit number \( ab \) can be expressed as:
\[ AB = 10a + b \]
Similarly, the two-digit number \( ba \) can be expressed as:
\[ BA = 10b + a \]

**Step 2: Substitute the value of \( b \).**
Given that \( b = a + 3 \), we can substitute this into the expressions for \( AB \) and \( BA \):
\[ AB = 10a + (a + 3) = 10a + a + 3 = 11a + 3 \]
\[ BA = 10(a + 3) + a = 10a + 30 + a = 11a + 30 \]

**Step 3: Compare the two expressions.**
Now we need to compare \( AB \) and \( BA \):
\[ AB = 11a + 3 \]
\[ BA = 11a + 30 \]

**Step 4: Determine the relationship.**
To find out which number is larger, we can compare:
\[ 11a + 3 \quad \text{(for } AB\text{)} \quad \text{and} \quad 11a + 30 \quad \text{(for } BA\text{)}. \]
Clearly, since \( 30 > 3 \), we have:
\[ 11a + 30 > 11a + 3 \]
Thus, \( BA > AB \).

**Step 5: Conclusion.**
Since \( BA \) is greater than \( AB \), we conclude that the positive two-digit number \( ba \) (Column B) is larger than the positive two-digit number \( ab \) (Column A).

**Final Answer:**
The correct option is that Column B is larger than Column A.

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