Simplification

What is VBODMAS?

When a long expression has $+, -, \times, \div$, brackets and bars, every calculator (and every examiner) follows one fixed order, called VBODMAS:

1. V — Vinculum: a bar over an expression, like $\overline{9-7}$, is cleared first.
2. B — Brackets: innermost first: $( )$ then $\{ \}$ then $[ ]$.
3. O — Of: 'of' means multiply, but it is done before ÷ and ×.
4. D and M: divisions and multiplications, strictly left to right.
5. A and S: additions and subtractions, again left to right.

The 'of' trap

$64 \div 4 \text{ of } 2$ is $64 \div 8 = 8$, not $(64 \div 4) \times 2 = 32$. Compare: $64 \div 4 \times 2 = 16 \times 2 = 32$ — no 'of', so left to right. Another form: $\frac{3}{4}$ of $48 \div 2$ = $(\frac{3}{4} \times 48) \div 2 = 18$, because 'of' groups $48$ with $\frac{3}{4}$ before any $\div$ outside.

Working through brackets

Always start from the innermost bracket. A minus sign in front of a bracket flips every sign inside: $25 - (10 - 6) = 25 - 10 + 6 = 21$. With three layers, clear $( )$, then $\{ \}$, then $[ ]$, one at a time. Do not try to open all brackets in one go.

Continued fractions

A fraction written inside a fraction, like $1 + \cfrac{1}{1 + \cfrac{1}{2}}$, is simplified from the bottom up:

• bottom: $1 + \frac{1}{2} = \frac{3}{2}$
• next: $\frac{1}{3/2} = \frac{2}{3}$
• top: $1 + \frac{2}{3} = \frac{5}{3}$.

Left-to-right chains

When only $\div$, $\times$ (and later $+, -$) appear, work strictly left to right: $60 \div 5 \times 3 = 12 \times 3 = 36$, not $60 \div 15 = 4$.

Two speed tools

• Digit-sum check (casting out 9s): replace each number by its digit sum (9 becomes 0), do the same operations on the digit sums, and compare with the digit sum of each option. It catches most slips. Example: $48 + 36 = 84$; digit sums $3 + 0 = 3$, and $84 \to 3$ ✓.
• Last-digit check: for $\times$ and $+$ chains, the option's last digit must match the last digit you get by working only with last digits. $7 \times 8 + 6 = 62$; last digits $6 + 6 = 12 \to 2$ ✓.

Approximate placement of values

Some questions ask "what comes in place of ?". Replace the ? by each option only after simplifying the fixed parts — do not solve with algebra when a check of two options is faster.

Quick revision

• Order: vinculum → brackets → of → ÷/× (L→R) → +/− (L→R).
• 'of' beats ÷ and ×.
• Minus before a bracket flips signs inside.
• Continued fractions: bottom-up.
• Verify with digit sums or last digits.