Which of the following arranges the fractions \(\displaystyle\frac{3}{5}, \frac{5}{8}, \frac{7}{11}\) and \(\displaystyle\frac{4}{7}\) in increasing order?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{4}{7} \lt \frac{3}{5} \lt \frac{5}{8} \lt \frac{7}{11}\)
For positive fractions, \(\frac{a}{b} \lt \frac{c}{d}\) exactly when \(ad \lt bc\). This cross-multiplication compares two fractions at a time.
\(\displaystyle \frac{4}{7} \text{ and } \frac{3}{5}: \; 4 \times 5 = 20 \lt 7 \times 3 = 21\)
\(\displaystyle \frac{3}{5} \text{ and } \frac{5}{8}: \; 3 \times 8 = 24 \lt 5 \times 5 = 25\)
\(\displaystyle \frac{5}{8} \text{ and } \frac{7}{11}: \; 5 \times 11 = 55 \lt 8 \times 7 = 56\)
The decimal values agree: \(0.571\ldots\), \(0.6\), \(0.625\) and \(0.636\ldots\). So the increasing order is \(\frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{7}{11}\).