Which of the following fractions lies between 2/3 and 3/5?

Concept: To compare fractions and check whether one lies between two others, convert every fraction to its decimal equivalent (divide numerator by denominator); decimal form makes the ordering unambiguous, unlike comparing fractions with different denominators directly.

1. Convert the two boundary fractions to decimals: 2/3 = 0.667 (recurring) and 3/5 = 0.6. The target fraction must lie strictly between 0.6 and 0.667.

2. Convert each option to a decimal: 2/5 = 0.4, 4/5 = 0.8, 8/9 ≈ 0.889, and 31/50 = 0.62.

3. Compare each decimal against the range (0.6, 0.667): 0.4, 0.8, and 0.889 all fall outside it, while 0.62 falls inside it.

Cross-check: comparing 31/50 with 3/5 by cross-multiplication gives 31 × 5 = 155 versus 3 × 50 = 150, so 31/50 > 3/5; comparing 31/50 with 2/3 gives 31 × 3 = 93 versus 2 × 50 = 100, so 31/50 < 2/3. This independently confirms the ordering without relying on decimal rounding.

So, 31/50 lies between 3/5 and 2/3.