`{:("Column A" , "a/2 is b% of 30, and a is c% of 50,b is positive","ColumnB"),(b,,c):}`

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 step by step, we will break down the information given in the question and derive the relationships between the variables \( a \), \( b \), and \( c \).

### Step 1: Understand the Relationships
We know from the problem statement:
1. \( \frac{a}{2} \) is \( b\% \) of 30.
2. \( a \) is \( c\% \) of 50.

### Step 2: Set Up the Equations
From the first statement:
\[ \frac{a}{2} = \frac{b}{100} \times 30 \]
This simplifies to:
\[ \frac{a}{2} = \frac{3b}{10} \]
Multiplying both sides by 2 gives:
\[ a = \frac{6b}{10} = \frac{3b}{5} \quad \text{(Equation 1)} \]

From the second statement:
\[ a = \frac{c}{100} \times 50 \]
This simplifies to:
\[ a = \frac{c}{2} \quad \text{(Equation 2)} \]

### Step 3: Equate the Two Expressions for \( a \)
Now, we can set the two expressions for \( a \) equal to each other:
\[ \frac{3b}{5} = \frac{c}{2} \]

### Step 4: Cross Multiply
Cross multiplying gives:
\[ 3b \cdot 2 = c \cdot 5 \]
This simplifies to:
\[ 6b = 5c \]

### Step 5: Solve for the Ratio \( \frac{b}{c} \)
Rearranging the equation gives:
\[ \frac{b}{c} = \frac{5}{6} \]

### Step 6: Analyze the Ratio
Since \( \frac{5}{6} < 1 \), it follows that:
\[ b < c \]

### Conclusion
From the analysis, we conclude that \( b \) is less than \( c \). Therefore, the answer to the comparison between Column A and Column B is that Column B is larger.

### Final Answer
Column B is larger.

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