`{:("Column A","The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + 7 is 4: 5", "Column B"),((x+14)/(y + 14),,5//6):}`

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 analyze the given ratios and find the values of \( x \) and \( y \) to compare the two columns.

### Step 1: Set up the ratios
We are given two ratios:
1. The ratio of \( x \) to \( y \) is \( 3:4 \), which can be expressed as:
\[ \frac{x}{y} = \frac{3}{4} \]
This implies:
\[ x = \frac{3}{4}y \]

2. The ratio of \( x + 7 \) to \( y + 7 \) is \( 4:5 \), which can be expressed as:
\[ \frac{x + 7}{y + 7} = \frac{4}{5} \]

### Step 2: Substitute \( x \) in the second ratio
Substituting \( x = \frac{3}{4}y \) into the second ratio:
\[ \frac{\frac{3}{4}y + 7}{y + 7} = \frac{4}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5\left(\frac{3}{4}y + 7\right) = 4(y + 7) \]

### Step 4: Expand both sides
Expanding both sides:
\[ \frac{15}{4}y + 35 = 4y + 28 \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 4:
\[ 15y + 140 = 16y + 112 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 15y - 16y = 112 - 140 \]
\[ -y = -28 \]
Thus,
\[ y = 28 \]

### Step 7: Find \( x \)
Now, substituting \( y = 28 \) back into the equation for \( x \):
\[ x = \frac{3}{4} \times 28 = 21 \]

### Step 8: Calculate Column A
Now we can calculate Column A:
\[ \text{Column A} = \frac{x + 14}{y + 14} = \frac{21 + 14}{28 + 14} = \frac{35}{42} \]
Simplifying \( \frac{35}{42} \):
\[ \frac{35}{42} = \frac{5}{6} \]

### Step 9: Compare with Column B
Column B is given as \( \frac{5}{6} \).

### Conclusion
Since both Column A and Column B are equal:
\[ \text{Column A} = \text{Column B} = \frac{5}{6} \]

Thus, the answer is that the two columns are equal.