Simplify: (5⁄6 + 3⁄4) ÷ (7⁄8−1⁄3)

Correct — A. Work each bracket first, then divide.

First bracket, LCD 12: 5⁄6 + 3⁄4 = 10⁄12 + 9⁄12 = 19⁄12
Second bracket, LCD 24: 7⁄8 − 1⁄3 = 21⁄24 − 8⁄24 = 13⁄24

Divide by multiplying by the reciprocal: 19⁄12 × 24⁄13
Cancel 24⁄12 = 2: (19 × 2)⁄13 = 38⁄13 → option (a)

Why the others are wrong

• (b)
  48⁄13 — 48⁄13 would need the first bracket to be 24⁄12 = 2. But 5⁄6 and 3⁄4 are each below 1, and their sum is 19⁄12, so the numerator is 19 × 2 = 38.
• (c)
  58⁄13 — 58⁄13 would need the first bracket to be 29⁄12. It is 10⁄12 + 9⁄12 = 19⁄12, which doubles to 38 over 13, not 58.
• (d)
  68⁄13 — 68⁄13 needs the first bracket to be 34⁄12, more than 2. Two fractions each below 1, like 5⁄6 and 3⁄4, cannot add to more than 2.

Brackets first, then the division. Each bracket needs its own common denominator: 12 for sixths and quarters, 24 for eighths and thirds.

Dividing by a fraction means multiplying by its reciprocal, so ÷ 13⁄24 becomes × 24⁄13. The 24 and the 12 then cancel to 2 before any large multiplication, leaving (19 × 2)⁄13.

Key facts

• a⁄b ÷ c⁄d = a⁄b × d⁄c.
• 5⁄6 + 3⁄4 = 19⁄12 and 7⁄8 − 1⁄3 = 13⁄24.
• The LCD of 6 and 4 is 12, and the LCD of 8 and 3 is 24.

Common traps

• Subtracting 7⁄8 − 1⁄3 as (7 − 1)⁄(8 − 3) = 6⁄5. Fractions subtract only over a common denominator.
• Inverting the wrong fraction: 19⁄12 ÷ 13⁄24 becomes 19⁄12 × 24⁄13, not 12⁄19 × 13⁄24.

No directly related past PYQ was found.