Question

Which among the following is the largest fraction ?

• A. \(1/2 \)
• B. \(1/21 \)
• C. \(1/5 \)
• D. \(1/9 \)

Hint:

For fractions with identical numerators, the fraction containing the smallest denominator will always hold the highest absolute value. No complex calculations are necessary!

Answer 1

The Correct Option is A

Concept: When comparing unit fractions (fractions where the numerator is equal to 1), the value of the fraction is inversely proportional to its denominator. In mathematical terms, for any positive integers \(a\) and \(b\): \[ \text{If } a \lt b, \text{ then } \frac{1}{a} \gt \frac{1}{b} \] Alternatively, we can convert each fraction into its equivalent decimal notation to carry out a direct numerical comparison.

Step 1: Analyzing fractions by looking at their denominators.
The given set of fractions consists of: \[ \frac{1}{2}, \, \frac{1}{21}, \, \frac{1}{5}, \, \frac{1}{9}, \, \frac{1}{15} \] Notice that all five fractions have an identical numerator of 1. According to fraction principles, when sharing a single unit into a certain number of parts, dividing it into fewer parts results in each individual slice being significantly larger. Let us arrange the denominators of the fractions in ascending order (from smallest to largest): \[ 2 \lt 5 \lt 9 \lt 15 \lt 21 \]

Step 2: Inverting the inequality to compare fraction values.
By taking the reciprocal of each denominator, the direction of the inequality signs reverses completely: \[ \frac{1}{2} \gt \frac{1}{5} \gt \frac{1}{9} \gt \frac{1}{15} \gt \frac{1}{21} \] From this ordered relation, we can clearly observe that the fraction with the smallest denominator, which is \(\frac{1}{2}\), yields the greatest total value.

Step 3: Verification via decimal conversion.
Let us convert each option into a decimal format to verify our conclusion: \[\begin{aligned} \frac{1}{2} &= 0.500 \\ \frac{1}{5} &= 0.200 \\ \frac{1}{9} &\approx 0.111 \\ \frac{1}{15} &\approx 0.066 \\ \frac{1}{21} &\approx 0.047 \end{aligned}\] Comparing the values: \(0.500 \gt 0.200 \gt 0.111 \gt 0.066 \gt 0.047\). This conclusively proves that \(\frac{1}{2}\) is the largest fraction among all choices.