The BODMAS Rule
BODMAS gives the order in which operations are solved: B - Brackets, O - Of (powers/orders), D - Division, M - Multiplication, A - Addition, S - Subtraction. Division and Multiplication rank EQUAL - do them left to right. Same for Addition and Subtraction. Bracket order: solve innermost first - ( ) then { } then [ ]. The word 'of' means multiply but is done before division (e.g. 1/2 of 8 = 4 first). Memory aid: 'Brackets Open, Divide Multiply, Add Subtract'. A frequent trap: students do addition before multiplication - always finish multiply/divide first. Sign rule: +x+ = +, -x- = +, +x- = -, -x+ = -.
Handling 'Of' and Fractions
'Of' is treated as multiplication but evaluated immediately after brackets and before plain division. Example: 1/3 of 12 / 2 = (1/3 x 12) / 2 = 4 / 2 = 2 - do the 'of' first. When simplifying fractions, convert mixed numbers to improper fractions, then apply BODMAS. To divide by a fraction, multiply by its reciprocal: a / (b/c) = a x c/b. For a 'fraction bar' that acts as a bracket, simplify the numerator and denominator separately before dividing. Memory tip: vinculum (bar over numbers like a line) is solved FIRST, even before brackets. Keep signs careful when removing brackets preceded by a minus sign - every inside term flips sign.
Worked Example: BODMAS Step-by-Step
"36 ÷ 4 × 3 + 8 − 5 × 2 — what is the answer?" In RPF Constable, SSC GD, and almost every banking prelims paper, one BODMAS expression sneaks in and a quarter of the candidates lose the mark. The maths is easy; the order is where everything goes wrong. This lesson works the full expression step by step so that the rule sinks in for life.
Definition: BODMAS is the agreed order of operations: Brackets → Orders (powers and roots) → Division and Multiplication (left to right) → Addition and Subtraction (left to right).
Definition: "Left to right" means within the D-M pair, you do whichever symbol appears first as you read the expression from left to right — not the one you find more familiar. The same rule applies to the A-S pair.
The question
Simplify: 36 ÷ 4 × 3 + 8 − 5 × 2
Apply BODMAS. There are no brackets, no powers — so we go straight to Division and Multiplication, left to right.
Step 1 — Division and Multiplication, strictly left to right
Read the expression from the left. The first D-or-M symbol you meet is the division 36 ÷ 4. Do it first:
36 ÷ 4 = 9
The expression becomes: 9 × 3 + 8 − 5 × 2
The next D-or-M symbol from the left is the multiplication 9 × 3:
9 × 3 = 27
The expression becomes: 27 + 8 − 5 × 2
Continue scanning. The next D-or-M symbol is 5 × 2:
5 × 2 = 10
The expression becomes: 27 + 8 − 10
All multiplications and divisions are now done.
Step 2 — Addition and Subtraction, strictly left to right
Again read from the left:
27 + 8 = 35
The expression becomes: 35 − 10
35 − 10 = 25
Conclusion: The value of 36 ÷ 4 × 3 + 8 − 5 × 2 is 25.
Why "left to right" is non-negotiable
Definition: Equal precedence. Division and multiplication share the same priority level; addition and subtraction share another, lower level. When two operators of equal precedence sit in the same expression, mathematicians long ago agreed to resolve them in reading order, i.e., left to right.
The catch is that BODMAS, as taught, lists "D" before "M". A surprising number of students read that as "always division first, then multiplication". It is not. The D and the M are tied; only their position in the expression decides which is done first. The same applies to A and S — addition is not automatically before subtraction.
The single most common mistake on this problem
Common misconception: Many candidates solve 36 ÷ 4 × 3 as 36 ÷ (4 × 3) = 36 ÷ 12 = 3, treating multiplication as automatically higher than division. The expression then collapses to 3 + 8 − 10 = 1 — completely wrong. The error is treating "BODMAS" as a fixed left-to-right list of priorities instead of grouping D-M as a single equal-precedence pair.
Another common error is to do 5 × 2 = 10 before 36 ÷ 4 = 9. That happens to give the same final answer here, but only by luck — change a sign and the trick fails. The safe habit is always left-to-right within the D-M tier.
A third error is treating subtraction as higher than addition. For 27 + 8 − 10, doing 8 − 10 = −2 first gives 27 − 2 = 25, which happens to be correct here, again by coincidence of sign. But the rule says: scan from the left, addition comes first, then subtraction. Build the habit before the exam, not in the exam.
Why it matters: BODMAS / order-of-operations questions appear in every quantitative aptitude paper — RPF Constable, RPF SI, SSC GD, SSC CHSL, SSC CGL, IBPS Clerk, SBI Clerk, Railway NTPC, state-level constabulary — usually as a quick one-mark question with negative marking. They are the easiest marks on the paper if you train the order, and a guaranteed loss if you do not.
Real-world example: When you key 36 ÷ 4 × 3 + 8 − 5 × 2 into a basic calculator at a railway counter (a simple, non-scientific calculator that processes left to right and does not respect order of operations), you may get a wrong number, because it adds 8 the moment you press +. The scientific calculator in your science classroom — and the one in your phone — does follow BODMAS and returns 25. The lesson: the order of operations is a convention humans agreed on; not every device follows it. The exam expects the BODMAS answer.
A second worked example to lock in the rule
Question: Simplify 48 ÷ 6 × 2 + 14 − 3 × 4.
Solution:
Step 1 (D-M left to right):
48 ÷ 6 = 8 → 8 × 2 + 14 − 3 × 4
8 × 2 = 16 → 16 + 14 − 3 × 4
3 × 4 = 12 → 16 + 14 − 12
Step 2 (A-S left to right):
16 + 14 = 30 → 30 − 12
30 − 12 = 18
Conclusion: 48 ÷ 6 × 2 + 14 − 3 × 4 = 18.
The procedure is identical to the main problem — proof that one drilled habit handles the entire genre.
| Stage | What to do | In our problem | Result |
|---|---|---|---|
| B | Solve brackets | None | — |
| O | Solve powers/roots | None | — |
| D & M | Left-to-right | 36 ÷ 4, 9 × 3, 5 × 2 | 27 + 8 − 10 |
| A & S | Left-to-right | 27 + 8, then 35 − 10 | 25 |
| Wrong path | Calculation | Wrong answer |
|---|---|---|
| Multiplication before division | 4 × 3 = 12, then 36 ÷ 12 = 3, then 3 + 8 − 10 = 1 | 1 (wrong) |
| Mixing the order of additions and subtractions | 27 + (8 − 10) = 25 — luck, not method | 25 (right by accident) |
| Correct BODMAS path | 9 × 3 = 27; 5 × 2 = 10; 27 + 8 − 10 = 25 | 25 |
- ✓- BODMAS gives the agreed order: Brackets, Orders, Division/Multiplication, Addition/Subtraction.
- ✓- D and M are at the same priority level; resolve them left to right in the expression.
- ✓- A and S are at the same lower level; resolve them left to right as well.
- ✓- 36 ÷ 4 × 3 = (36 ÷ 4) × 3 = 27, never 36 ÷ (4 × 3) = 3.
- ✓- Without brackets, you can never assume that the "later" operator is meant to bind tighter.
- ✓- The same rule covers any combination — division then multiplication, addition then subtraction, in any sequence.
- ✓- Train the eye to scan from left to right within each tier; speed comes from habit, not from cleverness.
"Before Old Dogs, Many Are Sent" — the classic acronym; remember to pair D with M and A with S, and resolve each pair left to right.
A second mantra for the exam hall: "Same-tier, same-direction — left first."
- ✓- 36 ÷ 4 × 3 + 8 − 5 × 2 = 25.
- ✓- After D and M (left to right): 27 + 8 − 10.
- ✓- After A and S (left to right): 35 − 10 = 25.
- ✓- The trap is treating D-M (or A-S) as a strict left-to-right priority list; they are equal-precedence pairs.
BODMAS & Simplification — Flashcards
Cover the answer, recall, then check. 12 cards on order of operations and speed identities.
Q1. What does BODMAS stand for, and what is the order?
A1. Brackets, Order (powers/roots), Division, Multiplication, Addition, Subtraction. Division & Multiplication rank equal (left to right); so do Addition & Subtraction.
Q2. Bracket priority order?
A2. Innermost first: bar (vinculum) → ( ) → { } → [ ].
Q3. Simplify 12 + 6 ÷ 2 × 3.
A3. ÷ and × first, left to right: 6÷2 = 3, 3×3 = 9, then 12 + 9 = 21.
Q4. Where does "of" sit in BODMAS?
A4. "of" is a multiplication done right after brackets/order, before ÷. e.g. 1/2 of 8 ÷ 2 = 4 ÷ 2 = 2.
Q5. Simplify 5 − [4 − {3 − (2 − 1)}].
A5. (2−1)=1 → {3−1}=2 → [4−2]=2 → 5−2 = 3.
Q6. √144 + √0.25 = ?
A6. 12 + 0.5 = 12.5.
Q7. Compute 43 × 37 using an identity.
A7. (40+3)(40−3) = 40² − 3² = 1600 − 9 = 1591.
Q8. Compute 105² quickly.
A8. (100+5)² = 10000 + 1000 + 25 = 11025.
Q9. Simplify 3/4 + 5/6 − 2/3.
A9. LCM 12: 9/12 + 10/12 − 8/12 = 11/12.
Q10. Write 0.333…, 0.666… and 0.1666… as fractions.
A10. 1/3, 2/3 and 1/6.
Q11. 15% of 240 by mental math?
A11. 10% = 24, 5% = 12 → 24 + 12 = 36.
Q12. Simplify 18 ÷ 2 of 3 (classic trap).
A12. "of" before ÷: 2 of 3 = 6, then 18 ÷ 6 = 3 — NOT 27.
BODMAS & Simplification — Summary
BODMAS and simplification is the warm-up of every quantitative paper, and RPF is no exception — 2–4 questions test nothing but disciplined order of operations and a few algebraic identities. These are guaranteed marks: no concept to "understand", only a rule to obey without slips.
The BODMAS order
Brackets → Order (powers & roots) → Division → Multiplication → Addition → Subtraction. Division/Multiplication share a rank (left-to-right); Addition/Subtraction share a rank (left-to-right). "of" is a multiplication done immediately after brackets.
Bracket nesting, innermost first: bar (vinculum) → ( ) → { } → [ ].
Handy identities (speed)
| Identity | Use |
|---|---|
| (a+b)² = a²+2ab+b² | 105² = 11025 |
| (a−b)² = a²−2ab+b² | 99² = 9801 |
| (a+b)(a−b) = a²−b² | 43×37 = 1591 |
| a³+b³ = (a+b)(a²−ab+b²) | factor/cancel |
| a³−b³ = (a−b)(a²+ab+b²) | factor/cancel |
Exam Tricks & Tips
- 🎯 Clear the innermost bracket first and rewrite the whole line each step — bracket slips cost more marks here than hard sums.
- 🎯 "of" outranks ÷ and ×: 18 ÷ 2 of 3 = 18 ÷ 6 = 3, not 27.
- 🎯 Use a²−b² for near-equal products: 96×104 = 100²−4² = 10000 − 16 = 9984.
- 🎯 Convert recurring decimals early: 0.333…=1/3, 0.1666…=1/6, 0.666…=2/3.
- 🎯 Memorise squares to 30 and cubes to 15 — most "simplify" sums hide a recall.
- ❌ Don't do addition before division just because it appears first — obey BODMAS order, not left-to-right reading order.
Expected exam pattern
Long single-line expressions with mixed ÷, ×, "of", brackets and a couple of square roots or fractions; or "find the value of ?" equations. Pure procedure — accuracy and speed decide.
Quick recap
Follow B-O-D-M-A-S strictly, clear innermost brackets first, treat "of" as priority multiplication, and deploy the three basic identities to shortcut big multiplications. Bank these easy marks fast.