Evaluate: 7 1⁄4 − [5⁄6 ÷ {1⁄3 − (1⁄2 × (3⁄4 − 1⁄4))}]

Correct — C. Work from the innermost bracket outward.

Innermost ( ): 3⁄4 − 1⁄4 = 1⁄2
Next ( ), multiply: 1⁄2 × 1⁄2 = 1⁄4
Curly { }: 1⁄3 − 1⁄4 = 4⁄12 − 3⁄12 = 1⁄12

Square [ ], divide: 5⁄6 ÷ 1⁄12 = 5⁄6 × 12 = 10
Subtract: 7 1⁄4 − 10 = −3 + 1⁄4 = −2 3⁄4 → option (c)

Why the others are wrong

• (a)
  −3 1⁄4 — −3 1⁄4 comes from writing −3 + 1⁄4 as a mixed number. But −3 1⁄4 means −(3 + 1⁄4) = −3.25, while −3 + 1⁄4 = −2.75.
• (b)
  3 1⁄4 — 3 1⁄4 is positive, yet the square bracket works out to 10, which is larger than 7 1⁄4, so 7 1⁄4 − 10 must be negative.
• (d)
  2 3⁄4 — Right size, wrong sign. 2 3⁄4 is 10 − 7 1⁄4, the subtraction done the wrong way round. The expression takes 10 away from 7 1⁄4.

Nested brackets are cleared from the inside out: here ( ) first, then { }, then [ ], with BODMAS inside each.

Two moves carry the marks. Dividing by a fraction means multiplying by its reciprocal, so ÷ 1⁄12 becomes × 12.

And a negative mixed number is one block: −2 3⁄4 means −(2 + 3⁄4). So 7 1⁄4 − 10 = −3 + 1⁄4 = −2 3⁄4, not −3 1⁄4.

Key facts

• Dividing by a fraction multiplies by its reciprocal: 5⁄6 ÷ 1⁄12 = 5⁄6 × 12 = 10.
• 1⁄n − 1⁄(n + 1) = 1⁄(n(n + 1)), so 1⁄3 − 1⁄4 = 1⁄12.
• A negative mixed number −a b⁄c means −(a + b⁄c).

Common traps

• Writing −3 + 1⁄4 as −3 1⁄4 instead of −2 3⁄4.
• Multiplying by 1⁄12 instead of dividing, which gives 5⁄72 and a final value just above 7.
• Working out 10 − 7 1⁄4 at the last step and dropping the minus sign.

No directly related past PYQ was found.