Arrange the fractions 5⁄9, 4⁄7, 3⁄5, and, 2⁄3 in ascending order.

Correct — B. Convert each fraction to a decimal (numerator ÷ denominator).

5⁄9 = 0.556
4⁄7 = 0.571
3⁄5 = 0.600
2⁄3 = 0.667

Order them smallest first: 5⁄9 < 4⁄7 < 3⁄5 < 2⁄3 → option (b)

Why the others are wrong

• (a)
  3⁄5, 4⁄7, 2⁄3, 5⁄9 — It opens with 3⁄5 = 0.6, but 4⁄7 = 0.571 is smaller, and it ends on 5⁄9, the smallest of the four, where the largest belongs.
• (c)
  2⁄3, 3⁄5, 4⁄7, 5⁄9 — This is the list largest first, the descending order, the exact reverse of what the question asks. 2⁄3 = 0.667 is the biggest, so it goes last.
• (d)
  4⁄7, 5⁄9, 2⁄3, 3⁄5 — It puts 4⁄7 before 5⁄9, but 4⁄7 = 0.571 is larger than 5⁄9 = 0.556. Placing 2⁄3 before 3⁄5 repeats the same mistake.

Every fraction here has the form n⁄(2n − 1), so each sits just above 1⁄2.

Subtract 1⁄2 and the excess is 1 ÷ (2 × denominator): 5⁄9 is 1⁄18 above, 4⁄7 is 1⁄14 above, 3⁄5 is 1⁄10 above and 2⁄3 is 1⁄6 above.

A smaller excess means a smaller fraction, so here the largest denominator gives the smallest fraction. The order follows the denominators 9, 7, 5, 3 with no division at all.

Key facts

• For positive fractions, a⁄b is smaller than c⁄d exactly when a × d is smaller than b × c.
• A fraction n⁄(2n − 1) exceeds 1⁄2 by 1⁄(2(2n − 1)), so it shrinks as n grows.
• Ascending order runs from smallest to largest.

Common traps

• Checking only the first two fractions of a list: 4⁄7 and 5⁄9 differ by less than 0.02, so a rough glance can swap them.
• Assuming a bigger denominator means a bigger fraction: 5⁄9 has the biggest denominator here and is the smallest.

No directly related past PYQ was found.