Finding the Least Fraction Among Given Options

We are given four fractions and asked to identify the least among them. The fractions are \(\frac{5}{11}\), \(\frac{7}{12}\), \(\frac{8}{13}\), and \(\frac{9}{17}\).

To find the least fraction, we can compare their values. A common method is to convert each fraction into its decimal equivalent. This allows for a direct comparison of their magnitudes.

Method: Converting Fractions to Decimals

To convert a fraction \(\frac{a}{b}\) to a decimal, we divide the numerator \(a\) by the denominator \(b\). Let's apply this to each fraction:

Calculating Decimal Values for Each Fraction

Fraction \(\frac{5}{11}\):

• Divide the numerator 5 by the denominator 11.
• \(5 \div 11 \approx 0.4545\) (rounded to four decimal places)

Fraction \(\frac{7}{12}\):

• Divide the numerator 7 by the denominator 12.
• \(7 \div 12 \approx 0.5833\) (rounded to four decimal places)

Fraction \(\frac{8}{13}\):

• Divide the numerator 8 by the denominator 13.
• \(8 \div 13 \approx 0.6154\) (rounded to four decimal places)

Fraction \(\frac{9}{17}\):

• Divide the numerator 9 by the denominator 17.
• \(9 \div 17 \approx 0.5294\) (rounded to four decimal places)

Comparing the Decimal Values

Now we have the approximate decimal values for all the fractions:

• \(\frac{5}{11} \approx 0.4545\)
• \(\frac{7}{12} \approx 0.5833\)
• \(\frac{8}{13} \approx 0.6154\)
• \(\frac{9}{17} \approx 0.5294\)

To find the least fraction, we need to find the smallest decimal value among these. Let's list them in order:

Comparing the values: \(0.4545, 0.5833, 0.6154, 0.5294\)

The smallest decimal value is \(0.4545\).

Conclusion: Identifying the Least Fraction

The decimal value \(0.4545\) corresponds to the fraction \(\frac{5}{11}\).

Therefore, \(\frac{5}{11}\) is the least fraction among the given options.

Revision Table: Comparing Fraction Values

From the table, it is clear that \(0.4545\) is the smallest decimal value, corresponding to \(\frac{5}{11}\).

Additional Information: Other Ways to Compare Fractions

While converting to decimals is useful, here are other methods to compare fractions:

• Common Denominator Method: Find the Least Common Multiple (LCM) of all denominators (11, 12, 13, 17). Convert each fraction to an equivalent fraction with this LCM as the new denominator. Compare the numerators; the fraction with the smallest numerator is the least. This can be complex if denominators are large or prime, like 11, 13, and 17.
• Cross-Multiplication Method: This is good for comparing two fractions at a time. To compare \(\frac{a}{b}\) and \(\frac{c}{d}\), calculate \(ad\) and \(bc\). If \(ad < bc\), then \(\frac{a}{b} < \frac{c}{d}\). If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\). If \(ad = bc\), then \(\frac{a}{b} = \frac{c}{d}\). You would need to perform multiple pairwise comparisons to find the least among several fractions.
• Comparing to a Benchmark: Compare fractions to a simple value like \(\frac{1}{2}\) or 1. For example, all given fractions are less than 1. \(\frac{5}{11}\) is close to \(\frac{1}{2}\) (which is \(\frac{5.5}{11}\)). \(\frac{7}{12}\) is also close to \(\frac{1}{2}\) (which is \(\frac{6}{12}\)). Comparing \(\frac{5}{11}\) and \(\frac{7}{12}\) using cross-multiplication: \(5 \times 12 = 60\) and \(11 \times 7 = 77\). Since \(60 < 77\), \(\frac{5}{11} < \frac{7}{12}\). This shows \(\frac{5}{11}\) is smaller than \(\frac{7}{12}\).

For comparing multiple fractions simultaneously, the decimal conversion method often provides a clear and direct way to see the values relative to each other.