Question

a and b are positive integers.
Column A: \(\frac{a}{b}\)
Column B: \(\frac{a+3}{b+3}\)

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

Hint:

Remember this rule: adding a positive number to both the numerator and denominator of a positive fraction moves the value of the fraction closer to 1. If the fraction was originally less than 1, it will increase. If it was originally greater than 1, it will decrease.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question asks how adding the same positive constant to the numerator and denominator of a fraction affects its value. The effect depends on the original relationship between the numerator and denominator.
Step 2: Detailed Explanation by Testing Cases:
Let's test different relationships between the positive integers \(a\) and \(b\).
Case 1: \(a \textless b\) (The fraction is less than 1) Let \(a=2\) and \(b=5\). - Column A: \(\frac{2}{5} = 0.4\) - Column B: \(\frac{2+3}{5+3} = \frac{5}{8} = 0.625\). In this case, Column B \textgreater Column A.
Case 2: \(a \textgreater b\) (The fraction is greater than 1) Let \(a=5\) and \(b=2\). - Column A: \(\frac{5}{2} = 2.5\) - Column B: \(\frac{5+3}{2+3} = \frac{8}{5} = 1.6\). In this case, Column A \textgreater Column B.
Case 3: \(a = b\) (The fraction is equal to 1) Let \(a=2\) and \(b=2\). - Column A: \(\frac{2}{2} = 1\) - Column B: \(\frac{2+3}{2+3} = \frac{5}{5} = 1\). In this case, Column A = Column B.
Step 3: Conclusion:
Since we have found scenarios where A \textgreater B, B \textgreater A, and A = B, the relationship cannot be determined from the information given.
Step 4: Algebraic Explanation (Optional):
To compare \(\frac{a}{b}\) and \(\frac{a+3}{b+3}\), we can subtract one from the other or use cross-multiplication (since b and b+3 are positive). Comparing \(a(b+3)\) with \(b(a+3)\). Comparing \(ab + 3a\) with \(ab + 3b\). Subtract \(ab\) from both sides: Compare \(3a\) with \(3b\). Compare \(a\) with \(b\). The comparison between the two columns depends entirely on the comparison between \(a\) and \(b\), which is not given.