Mathematical Operations

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Operator puzzles and BODMAS

⏱ 5 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

These questions give you a plain arithmetic line, or an equation with a missing number. The sums are easy. What is checked is the order of operations, called BODMAS. Do brackets first, then powers and 'of', then multiply and divide, then add and subtract.

What this topic tests

The numbers are small and the sums are easy. The examiner checks one thing: do you follow the order of operations without a slip?

Think of a shop bill. 3 pens at ₹10 each and 2 notebooks at ₹40 each cost $3 \times 10 + 2 \times 40 = 30 + 80 = 110$ rupees. Nobody adds 10 + 2 first. That habit is the whole topic.

BODMAS in simple words

BODMAS tells you which part to work out first.

Letter	Meaning	Example
B	Brackets	(6 + 2) is done first
O	Orders: powers, roots, "of"	$3^2$ , $\sqrt{16}$ , half of 10
D, M	Division and Multiplication	same rank
A, S	Addition and Subtraction	same rank

Example: 18 + 24 ÷ 6 × 3 − 5. First 24 ÷ 6 = 4, then 4 × 3 = 12. Now 18 + 12 − 5 = 25.

Same rank means left to right

Division is not before multiplication. Addition is not before subtraction. Same-rank signs are worked from left to right.

96 ÷ 8 × 4 = 12 × 4 = 48, not 96 ÷ 32 = 3.
60 − 12 + 8 = 48 + 8 = 56, not 60 − 20 = 40.

Watch: Examiners love to put ÷ before ×, or − before +. That one rule decides many answers.

The word "of" counts early

"Of" means multiply, but it is done before ÷. It sits with the O of BODMAS.

Example: 12 ÷ 3 of 4. First 3 of 4 = 12. Then 12 ÷ 12 = 1. Left to right would give 16 — that is the planted wrong answer.

Brackets inside brackets

Open brackets from the inside: first ( ), then { }, then [ ].

100 − [20 + {30 − (12 − 4) × 2}]. Inner: 12 − 4 = 8. Then 30 − 8 × 2 = 30 − 16 = 14. Then 20 + 14 = 34. Finally 100 − 34 = 66.

Rule: BODMAS works inside every bracket too. Do 8 × 2 before 30 − 8 × 2.

A method for long lines

Long lines look scary. They are only small steps in a row.

Circle every bracket and "of". Solve them first.
Circle every × and ÷. Solve them from left to right.
Rewrite the shorter line after each step.
Finish with + and −, from left to right.

Example: 40 − 6 × 3 + 20 ÷ 4. Circle 6 × 3 = 18 and 20 ÷ 4 = 5. The line is now 40 − 18 + 5 = 27.

Finding a missing number

One number is hidden: 18 + 6 × ? − 4 = 38. Undo the steps in reverse.

Move the loose terms: 6 × ? = 38 + 4 − 18 = 24.
Undo the multiply: ? = 24 ÷ 6 = 4.
Put 4 back: 18 + 24 − 4 = 38. It fits.

Tip: Always put your answer back. It takes five seconds and removes every doubt.

How wrong options are built

Three slips build almost every wrong option.

Left-to-right slip: 7 + 8 × 2 − 5 gives 25 left to right, but 7 + 16 − 5 = 18 is correct.
Sign slip: 45 − 5 × 6 + 9 is 45 − 30 + 9 = 24, not 45 − (30 + 9) = 6.
Forgotten term: the last − 5 or + 4 is left out.

Tip: Write the reduced line after each step, such as 18 + 12 − 5. Short lines prevent slips.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1common3 practice Q

Evaluate a mixed expression with BODMAS

How to spot it:

A plain expression with + − × ÷ and no brackets. Options include the left-to-right value.

Method

Circle every × and ÷ and work them out, left to right.
Replace each piece with its value.
Finish with + and −, left to right.
Match with the options.

Why it works:

× and ÷ bind the numbers next to them more tightly than + and −.

Try this

Find the value of 18 + 24 ÷ 6 × 3 − 5.

Show solution

24 ÷ 6 = 4, then 4 × 3 = 12.
Line: 18 + 12 − 5.
18 + 12 = 30, 30 − 5 = 25.

Answer

Type 2common3 practice Q

Same-rank operators: left to right

How to spot it:

A chain of only ÷ and × (or only + and −), where ÷ comes before × or − before +.

Method

Notice that every operator has the same rank.
Work strictly from left to right.
Keep the running value after each step.

Why it works:

BODMAS lists D and M together, and A and S together. The letter order is only a memory aid.

Try this

Find 96 ÷ 8 × 4.

Show solution

96 ÷ 8 = 12.
12 × 4 = 48.

Answer

Type 3common3 practice Q

Nested brackets

How to spot it:

The expression has ( ), { } and [ ] brackets.

Method

Solve the innermost ( ) first.
Then the { }, then the [ ].
Use BODMAS inside every bracket.
Finish outside the brackets.

Why it works:

Each bracket is a complete small expression whose value feeds the next one.

Try this

Find [48 ÷ {2 × (10 − 4)}] + 5.

Show solution

10 − 4 = 6.
2 × 6 = 12.
48 ÷ 12 = 4.
4 + 5 = 9.

Answer

Type 4common3 practice Q

Find the missing number in an equation

How to spot it:

An equation with a '?' in place of one number.

Method

Undo the + and − terms: move them to the other side.
Undo the × or ÷ around the '?'.
Put the value back and check both sides.

Why it works:

Undoing the steps in reverse keeps both sides equal at every move.

Try this

18 + 6 × ? − 4 = 38. Find ?.

Show solution

6 × ? = 38 + 4 − 18 = 24.
? = 24 ÷ 6 = 4.
Check: 18 + 6 × 4 − 4 = 38.

Answer

Type 5occasional

The word 'of' inside an expression

How to spot it:

A ÷ line containing the word 'of', like 12 ÷ 3 of 4.

\text{of} \equiv \times \text{, done before } \div

Method

Read 'of' as multiply.
Do the 'of' piece first, before any ÷.
Then finish with BODMAS as usual.

Why it works:

'Of' sits in the O of BODMAS, one rank above division.

Try this

Find 12 ÷ 3 of 4.

Show solution

3 of 4 = 3 × 4 = 12.
12 ÷ 12 = 1.
Left to right would give 16 — the planted answer.

Answer

Formula sheet

BODMAS order

B \to O \to D/M \to A/S

D and M share a rank and go left to right; so do A and S.

Missing number

? = \dfrac{\text{RHS} - \text{loose terms}}{\text{factor}}

Undo + and − first, then the multiply or divide around the ?.

Shortcuts that save time

⚡ Circle the × and ÷ first

Read the line once and circle every × and ÷. Work those pieces first. Then read the line again for + and −. Your eye lands on the high-rank work before any arithmetic.

Example

Find 7 + 8 × 2 − 5.

Show solution

Circled: $8 \times 2 = 16$ .
Line becomes $7 + 16 - 5$ .
Add and subtract left to right.

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Working strictly left to right when BODMAS applies.

Do × and ÷ first, then + and −, left to right within each rank.

Mistake 02

Doing × before a ÷ that comes earlier in the line.

÷ and × have equal rank. Go left to right.

Mistake 03

Opening brackets from the outside in.

Open the innermost ( ) first, then { }, then [ ].

Mistake 04

Treating 'of' like an ordinary multiply at its place.

'Of' belongs to the O of BODMAS and is done before ÷.

Mistake 05

Finding the missing number and moving on without checking.

Put the value back into the equation. Confirm both sides match.

Mistake 06

Dropping the last small term, like − 5 at the end.

After each step, rewrite the full reduced line before continuing.

Quick revision

Read this the night before the exam.

Order: Brackets, then Orders (powers, 'of'), then ÷ and ×, then + and −.
÷ and × have equal rank; + and − have equal rank. Same rank = left to right.
'Of' means multiply, but it is done before ÷.
Nested brackets: ( ) first, then { }, then [ ]; BODMAS inside each.
Missing number: undo the steps backwards, then put the value back.
The left-to-right value is the examiner's favourite wrong option.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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