Write these fractions in order of size, starting with the smallest. \(\frac{5}{9}\) \(\frac{7}{13}\) \(\frac{4}{7}\) \(\frac{6}{11}\)

These fractions have different numerators and different denominators, so no simple rule ranks them by inspection. Convert each to a decimal by dividing, which puts them all in a form that can be compared digit by digit.

• \( \frac{5}{9} = 5 \div 9 = 0.5555\ldots \)
• \( \frac{7}{13} = 7 \div 13 = 0.5384\ldots \)
• \( \frac{4}{7} = 4 \div 7 = 0.5714\ldots \)
• \( \frac{6}{11} = 6 \div 11 = 0.5454\ldots \)

Writing each fraction as a decimal is the method step [M1]

All four begin \( 0.5 \), so compare the hundredths: \( 0.53 \lt 0.54 \lt 0.55 \lt 0.57 \).

Written back as fractions and starting with the smallest, the order is \( \frac{7}{13} \), \( \frac{6}{11} \), \( \frac{5}{9} \), \( \frac{4}{7} \) [A1]

Every one of these fractions is a little more than a half, which is why the decimals agree to one decimal place; that is exactly why rounding too early is dangerous. Keep at least three decimal places when the values are this close. The misconception that the largest denominator gives the smallest fraction fails here: \( \frac{7}{13} \) has neither the largest nor the smallest denominator yet is the smallest fraction.