Finding the Largest Fraction: Step-by-Step Comparison

To find the largest fraction among the given options, we can compare them. One common method for comparing fractions is to convert each fraction into its decimal equivalent. This allows for a direct comparison of their values.

Let's convert each of the given fractions into decimal form:

Decimal Conversion of Fractions

• For \( \frac{7}{9} \): Divide 7 by 9. \( \frac{7}{9} \approx 0.777... \) (It's a repeating decimal)
• For \( \frac{11}{14} \): Divide 11 by 14. \( \frac{11}{14} \approx 0.7857... \)
• For \( \frac{3}{4} \): Divide 3 by 4. \( \frac{3}{4} = 0.75 \)
• For \( \frac{10}{13} \): Divide 10 by 13. \( \frac{10}{13} \approx 0.7692... \)

Comparing the Decimal Values

Now we have the approximate decimal values for each fraction:

• \( \frac{7}{9} \approx 0.777 \)
• \( \frac{11}{14} \approx 0.7857 \)
• \( \frac{3}{4} = 0.75 \)
• \( \frac{10}{13} \approx 0.7692 \)

Let's list these decimal values in ascending order to clearly see which one is the largest:

\( 0.75 < 0.7692 < 0.777... < 0.7857... \)

From the comparison, the largest decimal value is approximately 0.7857, which corresponds to the fraction \( \frac{11}{14} \).

Therefore, the largest fraction among the given options is \( \frac{11}{14} \).

Revision Table: Comparing Fractions for Size

Additional Information: Other Ways to Compare Fractions

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

• Finding a Common Denominator: Convert all fractions to equivalent fractions with the same denominator (the least common multiple of all denominators). Then, compare the numerators. The fraction with the largest numerator is the largest. For example, to compare \( \frac{3}{4} \) and \( \frac{2}{3} \), the common denominator is 12. \( \frac{3}{4} = \frac{9}{12} \) and \( \frac{2}{3} = \frac{8}{12} \). Since 9 > 8, \( \frac{3}{4} > \frac{2}{3} \).
• Cross-Multiplication: To compare two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), compare \( a \times d \) and \( b \times c \).
  • If \( a \times d > b \times c \), then \( \frac{a}{b} > \frac{c}{d} \).
  • If \( a \times d < b \times c \), then \( \frac{a}{b} < \frac{c}{d} \).
  • If \( a \times d = b \times c \), then \( \frac{a}{b} = \frac{c}{d} \).
  For example, comparing \( \frac{7}{9} \) and \( \frac{11}{14} \): \( 7 \times 14 = 98 \) and \( 9 \times 11 = 99 \). Since 98 < 99, \( \frac{7}{9} < \frac{11}{14} \). You can use this method to compare fractions two at a time.
• Comparing with a Benchmark: Compare fractions to a known value like \( \frac{1}{2} \) or 1. For instance, \( \frac{3}{4} \) is larger than \( \frac{1}{2} \) because \( 3 \times 2 > 4 \times 1 \).

Choosing the best method depends on the fractions you are comparing. Converting to decimals is often easiest when comparing multiple fractions directly, as done in this problem finding the largest fraction.