Question

The greatest number among the fractions \( \frac{1}{5}, \frac{1}{4}, \frac{1}{11}, \frac{1}{13} \) is:

• A. \( \frac{1}{5} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{11} \)
• D. \( \frac{1}{13} \)

Hint:

When numerators are equal, look directly for the smallest denominator to find the largest fraction. Think of it practically: sharing a single pizza among 4 people gives each person a larger slice than sharing it among 5, 11, or 13 people!

Answer 1

The Correct Option is B

Concept: When comparing unit fractions (fractions where the numerator is exactly 1), the size of the fraction is completely determined by its denominator.
• Inverse Relationship Rule: For fractions with identical positive numerators, the fraction with the smaller denominator holds the larger total value because a single unit is being divided into fewer pieces.
• Conversely, the fraction with the larger denominator will have a smaller net value.

Step 1: Analyzing and ordering the denominators.
Let's list out all denominators present in the set of given fractions: \[ \text{Denominators: } 5, \, 4, \, 11, \, 13 \] Arranging these positive integers in ascending order (from smallest to largest): \[ 4 < 5 < 11 < 13 \]

Step 2: Applying the reciprocal rule for fractions.
Since the denominators follow the order \(4 < 5 < 11 < 13\), taking the reciprocal reverses the inequality signs completely: \[ \frac{1}{4} > \frac{1}{5} > \frac{1}{11} > \frac{1}{13} \] From this clear ordered inequality sequence, it is visually apparent that the fraction \( \frac{1}{4} \) stands as the largest value among all elements in the set.

Step 3: Verification via decimal conversion.
Let us perform the explicit division for each option to convert them into standard decimal notation: \frac{1}{4} &= 0.250
\frac{1}{5} &= 0.200
\frac{1}{11} &\approx 0.091
\frac{1}{13} &\approx 0.076 Comparing the values: \(0.250 > 0.200 > 0.091 > 0.076\). This confirms that \( \frac{1}{4} \) is the greatest number, aligning with Option (B).