Evaluate the expression: (1⁄4 ÷ 1⁄2) ÷ (1⁄15 + 1 − 9⁄10)

Correct — C. Work each bracket, then divide.

First bracket: 1⁄4 ÷ 1⁄2 = 1⁄4 × 2 = 1⁄2

Second bracket, over the common denominator 30:
1⁄15 = 2⁄30, 1 = 30⁄30, 9⁄10 = 27⁄30
2⁄30 + 30⁄30 − 27⁄30 = 5⁄30 = 1⁄6

Divide: 1⁄2 ÷ 1⁄6 = 1⁄2 × 6 = 3 → option (c)

Why the others are wrong

• (a)
  5 — 5 comes from dropping the 1⁄15. The second bracket then becomes 1 − 9⁄10 = 1⁄10, and 1⁄2 ÷ 1⁄10 = 5. With 1⁄15 kept, the bracket is 1⁄6.
• (b)
  6 — 6 is 1 ÷ 1⁄6, which drops the first bracket. 1⁄4 ÷ 1⁄2 is 1⁄2, not 1, so the result is half of 6.
• (d)
  2 — 2 would need the second bracket to equal 1⁄4. In thirtieths it is 2 + 30 − 27 = 5, so 5⁄30 = 1⁄6, and 1⁄2 ÷ 1⁄6 = 3.

Dividing by a fraction means multiplying by its reciprocal: 1⁄4 ÷ 1⁄2 = 1⁄4 × 2⁄1. Brackets come first, so each bracket is reduced to a single fraction before the outer division.

For a bracket that mixes denominators, rewrite every term over the LCM of the denominators (the LCM of 15 and 10 is 30, and 1 = 30⁄30), then add and subtract the numerators.

Key facts

• a⁄b ÷ c⁄d = a⁄b × d⁄c = ad⁄bc.
• LCM(15, 10) = 30, so 1⁄15 = 2⁄30 and 9⁄10 = 27⁄30.
• Dividing by a fraction below 1 gives a result larger than the number divided: 1⁄2 ÷ 1⁄6 = 3.

Common traps

• Inverting the wrong fraction: 1⁄4 ÷ 1⁄2 is 1⁄4 × 2 = 1⁄2, while flipping the first term instead gives 2.
• Losing the whole number 1 inside the second bracket: it must become 30⁄30 before the fractions are combined.

No directly related past PYQ was found.