Simplify: \12/33 ÷ 12/2\ ÷ (3/7 × 14/9 + 3/11) + 2/8 ÷ 31/25 of 25/8

Question

Simplify:

\12/33 ÷ 12/2\ ÷ (3/7 × 14/9 + 3/11) + 2/8 ÷ 31/25 of 25/8

1. A. 4/31 (Correct answer)
2. B. 5/36
3. C. 7/38
4. D. 5/24

Correct Answer

Option A — 4/31

Detailed Solution & Explanation

The correct answer is $\dfrac{4}{31}$.

Shortcut Trick

• Resolve the two groupings separately; each collapses to a small fraction.
• Braces: $\dfrac{12}{33} \div \dfrac{12}{2} = \dfrac{12}{33} \times \dfrac{2}{12} = \dfrac{2}{33}$
• Parentheses: $\dfrac{3}{7} \times \dfrac{14}{9} + \dfrac{3}{11} = \dfrac{2}{3} + \dfrac{3}{11} = \dfrac{22 + 9}{33} = \dfrac{31}{33}$
• Dividing: $\dfrac{2}{33} \div \dfrac{31}{33} = \dfrac{2}{31}$
• "of" binds first: $\dfrac{31}{25} \text{ of } \dfrac{25}{8} = \dfrac{31}{8}$, so $\dfrac{2}{8} \div \dfrac{31}{8} = \dfrac{2}{31}$
• Total $= \dfrac{2}{31} + \dfrac{2}{31} = \dfrac{\textbf{4}}{\textbf{31}}$

Alternate Method

• Notice the symmetry: both halves of the expression reduce to $\dfrac{2}{31}$. Spotting that after computing the first half lets you double it rather than working the second through.
• The design is deliberate — the 33s cancel in the first half and the 8s in the second.

Additional Information

• "of" is treated as multiplication but binds more tightly than division in this convention, so $\frac{2}{8} \div \frac{31}{25}$ of $\frac{25}{8}$ means $\frac{2}{8} \div \left(\frac{31}{25} \times \frac{25}{8}\right)$, not $\left(\frac{2}{8} \div \frac{31}{25}\right) \times \frac{25}{8}$. Reading it the second way gives a different answer entirely.
• Bracket order is braces { } → parentheses ( ) → the rest, worked inside out.
• Cancel before multiplying at every step; carrying $\frac{45}{60}$-style products through a long chain is where arithmetic slips creep in.