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{:("Column A",x//15 > y//25,"Column B),(...

`{:("Column A",x//15 > y//25,"Column B),(6y + 5x,,10x + 3y):}`

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 compare the two columns given the inequality condition \( \frac{x}{15} > \frac{y}{25} \). ### Step-by-Step Solution 1. **Understanding the Inequality**: We start with the condition given: \[ \frac{x}{15} > \frac{y}{25} \] This can be rewritten as: \[ x > \frac{15}{25}y \quad \text{or} \quad x > \frac{3}{5}y \] **Hint**: Rewrite the inequality in a simpler form to understand the relationship between \(x\) and \(y\). 2. **Setting Up the Columns**: We have: - Column A: \(6y + 5x\) - Column B: \(10x + 3y\) 3. **Assuming Column A is Greater**: Let's assume \(6y + 5x > 10x + 3y\). Rearranging gives: \[ 6y - 3y > 10x - 5x \] Simplifying this leads to: \[ 3y > 5x \quad \text{or} \quad y > \frac{5}{3}x \] **Hint**: When comparing two expressions, rearranging can help isolate the variables. 4. **Comparing with the Given Condition**: We now have two inequalities: - From our assumption: \(y > \frac{5}{3}x\) - From the original condition: \(x > \frac{3}{5}y\) To compare these, rewrite the condition \(x > \frac{3}{5}y\) as: \[ y < \frac{5}{3}x \] **Hint**: Make sure to keep track of the direction of the inequalities when manipulating them. 5. **Contradiction**: We see that \(y > \frac{5}{3}x\) contradicts \(y < \frac{5}{3}x\). Therefore, our assumption that Column A is greater cannot be true. **Hint**: If you find a contradiction, it means your initial assumption was wrong. 6. **Assuming Column B is Greater**: Now, let's assume \(10x + 3y > 6y + 5x\). Rearranging gives: \[ 10x - 5x > 6y - 3y \] Simplifying this leads to: \[ 5x > 3y \quad \text{or} \quad x > \frac{3}{5}y \] **Hint**: Similar to before, rearranging can help clarify the relationship. 7. **Conclusion**: The condition \(x > \frac{3}{5}y\) is consistent with the original inequality \(x > \frac{3}{5}y\). Thus, we conclude that Column B is indeed greater than Column A. Therefore, the answer is that **Column B is larger**. ### Final Answer: **Column B is larger (Option 2).**
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