Simplify the following expression. 133 ÷ 19 × 3 − 48 × 2 ÷ 24 + 841 ÷ [ { 4̅5̅ ̅−̅ ̅1̅6̅ } ]

Correct — C. The stem is an image reading 133 ÷ 19 × 3 − 48 × 2 ÷ 24 + 841 ÷ [ { 45 − 16 } ], with a bar drawn over 45 − 16.

That bar is a vinculum, and it is resolved before any bracket:
45 − 16 = 29, so [ { 29 } ] = 29

Now left to right within each rank:
133 ÷ 19 = 7, then 7 × 3 = 21
48 × 2 = 96, then 96 ÷ 24 = 4
841 ÷ 29 = 29

21 − 4 + 29 = 46, which is option (c).

Why the others are wrong

• (a)
  42 — Option (a) 42 does not follow from the three terms this line yields. They are 133 ÷ 19 × 3 = 21, 48 × 2 ÷ 24 = 4 and 841 ÷ 29 = 29, and 21 − 4 + 29 is 46.
• (b)
  12 — Option (b) 12 is the size of 21 − (4 + 29) = −12, what you get by subtracting the whole tail instead of only the 4. Addition and subtraction run left to right.
• (d)
  21 — Option (d) 21 is the first term on its own. The rest does not cancel: −4 + 29 is +25, not zero.

Simplification lines are graded on order, not on arithmetic. The order is V–B–O–D–M–A–S: vinculum, brackets, "of", division and multiplication, then addition and subtraction.

The vinculum is the bar drawn over 45 − 16. It outranks the brackets around it, so 45 − 16 = 29 happens first and the [ { } ] are left holding a single number.

Inside the division-multiplication rank, and again inside the addition-subtraction rank, work left to right. That is why 133 ÷ 19 × 3 is 21 and not 133 ÷ 57.

Key facts

• A vinculum, the bar drawn over part of an expression, is resolved before any bracket.
• Division and multiplication share a rank and are done left to right, as are addition and subtraction.
• 841 = 29 × 29, so 841 ÷ 29 = 29.
• The three terms of this line are 21, 4 and 29, giving 21 − 4 + 29 = 46.

Common traps

• Opening the square bracket before the bar and dividing 841 by 45.
• Treating everything after the minus sign as one subtracted block, which gives −12.
• Reading 133 ÷ 19 × 3 as 133 ÷ (19 × 3) = 133 ÷ 57.

No directly related past PYQ was found.