BODMAS Rule: Order of Operations with Worked Examples

Why the Order of Operations Matters

What is 8 − 2 × 3? If you work from left to right you get 6 × 3 = 18. If you multiply first you get 8 − 6 = 2. Only one of these can be right, and mathematicians agreed long ago on a fixed order so that every expression has exactly one value. In Nigerian and British schools this agreed order is remembered with the word BODMAS: Brackets, Orders (powers and roots), Division and Multiplication, Addition and Subtraction. The correct answer is therefore 2. This tutorial explains each step of BODMAS, the common traps (especially with division, subtraction and negative numbers), how directed numbers behave, and how calculators apply the same rules, with worked examples of the kind set in WAEC, NECO and JAMB.

The BODMAS Rule Step by Step

Work through an expression in this order:

B – Brackets. Work out anything inside brackets first. With brackets inside brackets, start with the innermost pair.
O – Orders. Powers (indices) and roots, such as 3², 2³ and √16.
D and M – Division and Multiplication. These have equal rank. Do them in the order they appear, from left to right.
A and S – Addition and Subtraction. These also have equal rank, worked from left to right.

Some books write BIDMAS (I for Indices) or, in America, PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). They are the same rule. PEMDAS lists multiplication before division, but that does not mean multiplication comes first: the two are always done together, left to right.

The Most Common Mistake: Left to Right

Because the D comes before the M in BODMAS, many students think division is always done before multiplication. It is not. Consider 24 ÷ 4 × 2.

Correct (left to right): 24 ÷ 4 = 6, then 6 × 2 = 12.
Wrong (multiplying first): 4 × 2 = 8, then 24 ÷ 8 = 3.

The same applies to subtraction and addition. In 10 − 3 + 2, work left to right: 10 − 3 = 7, then 7 + 2 = 9. Adding first (3 + 2 = 5, then 10 − 5 = 5) is wrong. A helpful way to see this is to think of subtraction as adding a negative number: 10 − 3 + 2 = 10 + (−3) + 2 = 9, and addition can be done in any order.

Worked Examples Using BODMAS

Example 1. Evaluate 5 + 3 × 4.

Multiplication comes before addition: 3 × 4 = 12, then 5 + 12 = 17.

Example 2. Evaluate (5 + 3) × 4.

Brackets first: 5 + 3 = 8, then 8 × 4 = 32. Brackets are how you override the normal order.

Example 3. Evaluate 18 ÷ 3² + 4 × (7 − 5).

Brackets: 7 − 5 = 2, giving 18 ÷ 3² + 4 × 2.
Orders: 3² = 9, giving 18 ÷ 9 + 4 × 2.
Division and multiplication: 18 ÷ 9 = 2 and 4 × 2 = 8, giving 2 + 8.
Addition: 10.

Example 4. Evaluate 2 × [15 − (3 + 2)²] ÷ 4.

Innermost brackets: 3 + 2 = 5.
Power inside the square brackets: 5² = 25.
Square brackets: 15 − 25 = −10.
Left to right: 2 × (−10) = −20, then −20 ÷ 4 = −5.

Example 5. Evaluate (12 + 8) ÷ (6 − 2) + 3 × 2³.

Brackets: 20 ÷ 4 + 3 × 2³. Orders: 2³ = 8. Then 20 ÷ 4 = 5 and 3 × 8 = 24. Finally 5 + 24 = 29.

Fraction Bars Act Like Brackets

A fraction bar groups everything above it and everything below it. In (6 + 9) ÷ (2 + 3) written as a fraction with 6 + 9 on top and 2 + 3 underneath, you must work out the top and bottom separately first: 15 ÷ 5 = 3. If you typed 6 + 9 ÷ 2 + 3 into a calculator without brackets, it would give 6 + 4.5 + 3 = 13.5, which is a different expression.

The same is true of a square root sign: √(9 + 16) = √25 = 5, while √9 + 16 = 3 + 16 = 19. The bar of the root sign covers everything under it, just like a bracket.

Directed Numbers (Positive and Negative Numbers)

Many BODMAS questions include negative numbers, so you need the rules for directed numbers.

Adding and subtracting:

Adding a negative is the same as subtracting: 7 + (−3) = 7 − 3 = 4.
Subtracting a negative is the same as adding: 7 − (−3) = 7 + 3 = 10.
On a number line, adding moves right and subtracting moves left; a negative sign reverses the direction.

Multiplying and dividing:

Same signs give a positive answer: (−4) × (−5) = 20 and (−12) ÷ (−3) = 4.
Different signs give a negative answer: (−4) × 5 = −20 and 12 ÷ (−3) = −4.
With several factors, count the negative signs: an even number gives a positive result, an odd number gives a negative result. (−1) × (−2) × (−3) = −6 because there are three negatives.

Powers of Negative Numbers: −3² or (−3)²?

This is a classic exam trap. The power applies only to what is immediately in front of it.

(−3)² means (−3) × (−3) = 9.
−3² means −(3²) = −9. The power is done first (O comes before the subtraction or negation), and the minus sign is applied afterwards.

Similarly (−2)³ = −8, and (−2)⁴ = 16: an even power of a negative number is positive, an odd power is negative.

Example 6. Evaluate −2² + (−3)² − 4 × (−1).

Orders: −2² = −4 and (−3)² = 9. Multiplication: 4 × (−1) = −4. So the expression is −4 + 9 − (−4) = −4 + 9 + 4 = 9.

Order of Operations with Fractions and Decimals

The same rules apply to fractions. Evaluate 1/2 + 1/3 × 3/4.

Multiplication first: 1/3 × 3/4 = 3/12 = 1/4. Then 1/2 + 1/4 = 2/4 + 1/4 = 3/4. Adding first would give 5/6 × 3/4 = 15/24 = 5/8, which is wrong.

A common WAEC-style question is "of" in fractions, such as 2/3 of 1½ + 1/4. The word "of" means multiply, and it is done with the multiplications: 2/3 × 3/2 = 1, then 1 + 1/4 = 1¼.

How Calculators Use BODMAS

Scientific calculators follow the order of operations automatically. Typing 5 + 3 × 4 gives 17, not 32. However, you must add brackets yourself wherever the written expression has a fraction bar, a long root sign or an exponent that covers several terms.

To work out (4 + 6) ÷ (7 − 2), type the brackets: the calculator gives 2. Without them it computes 4 + 6 ÷ 7 − 2, about 2.857.
To square a negative number, type (−3)² with brackets. Typing −3² gives −9.
For 2 to the power (n + 1), put the whole power in brackets.

The same rules are built into programming languages. In Python, for example, 8 - 2 * 3 is 2, and -3 ** 2 is −9, exactly as BODMAS predicts.

Practice Questions

Evaluate 20 − 4 × 3 + 2. Answer: 20 − 12 + 2 = 10.
Evaluate 36 ÷ 6 × 3. Answer: 6 × 3 = 18.
Evaluate (2 + 3)² − 2 × 3². Answer: 25 − 18 = 7.
Evaluate −5 − (−8) + 3 × (−2). Answer: −5 + 8 − 6 = −3.
Evaluate 3 + 4 × [10 − 2 × (6 − 4)]. Answer: 6 − 4 = 2; 10 − 4 = 6; 4 × 6 = 24; 3 + 24 = 27.
Evaluate (−2)³ − (−2)². Answer: −8 − 4 = −12.

Summary

BODMAS gives the order of operations: Brackets, Orders, Division and Multiplication, Addition and Subtraction.
Division and multiplication have equal rank and are done left to right; so are addition and subtraction.
Fraction bars and root signs act like brackets.
For directed numbers, equal signs give a positive product or quotient and different signs give a negative one.
(−3)² = 9 but −3² = −9, because the power is applied before the minus sign.
Calculators follow the same rules, so add brackets wherever the written expression has groupings.