Question

\(x>0\)
Column A: \(\frac{590 + x}{800}\)
Column B: \(\frac{600 + x}{790}\)

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

Another quick way to solve this is by inspection. In Column B, you are adding a larger number (600 vs 590) to \(x\), and then dividing by a smaller number (790 vs 800). Both of these changes make the fraction in Column B larger than the fraction in Column A. When the numerator is larger and the denominator is smaller, the resulting fraction is always greater (for positive numbers).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to compare two algebraic fractions where the variable \(x\) is positive. There are several methods, including cross-multiplication, plugging in a number, or analyzing the structure of the fractions.
Step 2: Key Formula or Approach:
The most rigorous method is cross-multiplication. Since the denominators (800 and 790) are positive, the direction of the inequality will not change. We will compare \((590 + x) \times 790\) with \((600 + x) \times 800\).
Step 3: Detailed Explanation:
Let's perform the cross-multiplication.
Left Side (from Column A):
\[ (590 + x) \times 790 = 590 \times 790 + 790x = 466100 + 790x \]
Right Side (from Column B):
\[ (600 + x) \times 800 = 600 \times 800 + 800x = 480000 + 800x \]
Comparison:
We are now comparing \(466100 + 790x\) with \(480000 + 800x\).
Let's subtract \(466100\) from both sides and \(790x\) from both sides to simplify the comparison.
We compare \(0\) with \((480000 - 466100) + (800x - 790x)\).
\[ 0 \quad \text{vs.} \quad 13900 + 10x \]
We are given that \(x>0\). This means \(10x\) is a positive number.
Therefore, \(13900 + 10x\) will always be a positive number greater than 13900.
Since \(13900 + 10x>0\), the right side of our comparison is always larger. The right side corresponds to Column B.
Step 4: Final Answer:
The quantity in Column B is greater.