Order of Operations and BODMAS

🎯 By the end of this lesson...

You should be able to:

Explain why calculations must follow an agreed order.
State what each letter in BODMAS represents.
Evaluate calculations containing brackets and powers.
Apply division and multiplication from left to right.
Apply addition and subtraction from left to right.
Use BODMAS with positive and negative numbers.
Check whether an answer is reasonable.

🤔 Think About This...

Two students are asked to calculate:

4 + 3 × 2

The first student works from left to right:

4 + 3 = 7

7 × 2 = 14

The second student multiplies first:

3 × 2 = 6

4 + 6 = 10

They obtained different answers from the same calculation.

To ensure that everyone obtains the same answer, mathematics uses an agreed order called the order of operations.

1️⃣ What Is the Order of Operations?

A calculation may contain several different operations, such as:

brackets;
powers;
division;
multiplication;
addition;
subtraction.

We cannot simply complete them in whichever order we prefer.

The order of operations tells us which part of a calculation must be completed first.

2️⃣ BODMAS

The word BODMAS helps us remember the correct order.

B

Brackets

Complete calculations inside brackets first.

O

Orders

Calculate powers and roots.

D

Division

Complete division from left to right.

M

Multiplication

Complete multiplication from left to right.

A

Addition

Complete addition from left to right.

S

Subtraction

Complete subtraction from left to right.

Orders means powers and roots, such as 3², 2³ and √16.

⚠ An Important BODMAS Rule

Division and multiplication have the same level of importance.

When both appear, work from left to right.

24 ÷ 6 × 2

Begin with the operation that appears first from the left:

24 ÷ 6 = 4

4 × 2 = 8

24 ÷ 6 × 2 = 8

Do not always complete multiplication before division merely because M appears after D in BODMAS.

Division and multiplication are completed from left to right.

➕ Addition and Subtraction

Addition and subtraction also have the same level of importance.

When both appear, work from left to right.

Example 1

Calculate:

18 − 7 + 3

Work from left to right:

18 − 7 = 11

11 + 3 = 14

18 − 7 + 3 = 14

🧠 The StudyNest BODMAS Method

Step 1: Look for brackets.

Complete the calculation inside each pair of brackets.

Step 2: Look for powers and roots.

Calculate orders such as squares, cubes and square roots.

Step 3: Look for division and multiplication.

Complete them from left to right.

Step 4: Look for addition and subtraction.

Complete them from left to right.

Step 5: Check your work.

Make sure that you completed every step in the correct order.

Do one clear step at a time. Rewriting the calculation after every step reduces careless mistakes.

3️⃣ Multiplication Before Addition

Example 2

Calculate:

6 + 4 × 5

There are no brackets or powers.

Multiplication comes before addition.

4 × 5 = 20

Rewrite the remaining calculation:

6 + 20

6 + 4 × 5 = 26

Do not calculate 6 + 4 first.

The correct answer is 26, not 50.

4️⃣ Brackets First

Example 3

Calculate:

3 × (7 − 2)

Complete the brackets first:

7 − 2 = 5

The calculation becomes:

3 × 5

3 × (7 − 2) = 15

📦

Why brackets matter

Suppose three learners each receive a pack containing seven pencils, but two pencils from each pack are damaged.

The usable pencils are represented by:

3 × (7 − 2)

The brackets show that we must first find the number of usable pencils in one pack.

5️⃣ Powers Before Multiplication

Example 4

Calculate:

5 + 2 × 3²

Calculate the power first:

3² = 9

The calculation becomes:

5 + 2 × 9

Multiply:

2 × 9 = 18

Add:

5 + 18 = 23

5 + 2 × 3² = 23

6️⃣ Roots Within Calculations

Example 5

Calculate:

20 − √16 × 3

Calculate the root first:

√16 = 4

The calculation becomes:

20 − 4 × 3

Multiply:

4 × 3 = 12

Subtract:

20 − 12 = 8

20 − √16 × 3 = 8

7️⃣ More Than One Pair of Brackets

Example 6

Calculate:

(8 − 3) × (4 + 2)

Complete each pair of brackets:

8 − 3 = 5

4 + 2 = 6

Multiply the results:

5 × 6 = 30

(8 − 3) × (4 + 2) = 30

8️⃣ Brackets Inside Brackets

Sometimes one pair of brackets appears inside another.

Begin with the innermost brackets and work outwards.

Example 7

Calculate:

2 × [5 + (7 − 3)]

Begin with the innermost brackets:

7 − 3 = 4

The calculation becomes:

2 × [5 + 4]

Complete the remaining brackets:

5 + 4 = 9

Multiply:

2 × 9 = 18

2 × [5 + (7 − 3)] = 18

Square brackets and ordinary brackets both group parts of a calculation. Work from the innermost grouping symbols outwards.

9️⃣ BODMAS with Directed Numbers

Example 8

Calculate:

−6 + 4 × 3

Multiply first:

4 × 3 = 12

The calculation becomes:

−6 + 12

−6 + 4 × 3 = 6

Example 9

Calculate:

5 − (−3) × 2

Multiply first:

(−3) × 2 = −6

The calculation becomes:

5 − (−6)

Subtracting a negative is the same as adding:

5 + 6 = 11

5 − (−3) × 2 = 11

🌍 Real-Life Example: A Shopping Bill

Example 10

A family buys:

3 loaves of bread costing US$2 each;
2 cartons of milk costing US$3 each.

They use a US$4 discount voucher.

The total cost is represented by:

3 × 2 + 2 × 3 − 4

Complete the multiplications first:

3 × 2 = 6

2 × 3 = 6

The calculation becomes:

6 + 6 − 4

Work from left to right:

12 − 4 = 8

The family pays US$8.

🌍 Real-Life Example: Football Points

Example 11

A football team receives:

3 points for each of 5 wins;
1 point for each of 2 draws;
a deduction of 4 points.

The total is:

3 × 5 + 1 × 2 − 4

Multiply first:

15 + 2 − 4

Work from left to right:

17 − 4 = 13

The team has 13 points.

🔎 Comparing Similar Calculations

Brackets can completely change an answer.

Without brackets

4 + 3 × 2

4 + 6 = 10

With brackets

(4 + 3) × 2

7 × 2 = 14

The brackets tell us that 4 + 3 must be completed before multiplying.

🎯 Exam Tips

Write one clear line for every stage of your calculation. This makes your method easier to follow and helps you notice mistakes.

Do not type the entire expression into a calculator without checking that the brackets and powers have been entered correctly.

When division and multiplication appear together, work from left to right. Do the same for addition and subtraction.

⚠ Common Mistakes

Mistake 1:

Working from left to right without first checking BODMAS.

Mistake 2:

Adding before multiplying.

In 4 + 3 × 2, multiplication must be completed first.

Mistake 3:

Believing multiplication must always come before division.

Complete division and multiplication from left to right.

Mistake 4:

Believing addition must always come before subtraction.

Complete addition and subtraction from left to right.

Mistake 5:

Ignoring brackets or completing only part of a bracketed calculation.

Mistake 6:

Treating 3² as 3 × 2.

Remember that 3² means 3 × 3.

Mistake 7:

Losing a negative sign while rewriting the calculation.

🎮 Your Turn!

Use the correct order of operations. Show each important step.

7 + 3 × 4
20 − 12 ÷ 3
18 ÷ 3 × 2
15 − 8 + 6
4 × (9 − 3)
(5 + 7) ÷ 3
2 + 3² × 4
30 − √25 × 4
(6 − 2) × (3 + 5)
3 × [4 + (8 − 6)]
−5 + 4 × 3
8 − (−2) × 5
36 ÷ 6 × 3 − 4
5 + 2 × (7 − 3)²
A customer buys 4 exercise books costing US$3 each and 2 pens costing US$1 each. A US$2 discount is deducted. Write and evaluate an expression for the total cost.

✅ Show Answers

7 + 12 = 19
20 − 4 = 16
6 × 2 = 12
7 + 6 = 13
4 × 6 = 24
12 ÷ 3 = 4
2 + 9 × 4 = 2 + 36 = 38
30 − 5 × 4 = 30 − 20 = 10
4 × 8 = 32
3 × [4 + 2] = 3 × 6 = 18
−5 + 12 = 7
8 − (−10) = 8 + 10 = 18
6 × 3 − 4 = 18 − 4 = 14
5 + 2 × 4²

= 5 + 2 × 16

= 5 + 32

= 37
4 × 3 + 2 × 1 − 2

= 12 + 2 − 2

= US$12

🧠 Remember This

BODMAS gives everyone the same answer.

Complete brackets first.

Then calculate powers and roots.

Complete division and multiplication from left to right.

Complete addition and subtraction from left to right.

Rewrite the calculation after every important step.

🎯 Before Moving On...

✅ Can I explain why the order of operations is necessary?
✅ Do I know what every letter in BODMAS represents?
✅ Can I complete brackets before other operations?
✅ Can I evaluate powers and roots correctly?
✅ Can I apply division and multiplication from left to right?
✅ Can I apply addition and subtraction from left to right?
✅ Can I use BODMAS with directed numbers?

🎉 Well Done!

You can now evaluate calculations containing several operations in the correct order.

Remember:

Brackets → Orders → Division and Multiplication → Addition and Subtraction

📈 Your Progress

Lesson 5 of 17

Real Numbers Progress 29%

Order of Operations

🎬 Watch the order unfold

🎯 By the end of this lesson...

🤔 Think About This...

1️⃣ What Is the Order of Operations?

2️⃣ BODMAS

Brackets

Orders

Division

Multiplication

Addition

Subtraction

⚠ An Important BODMAS Rule

➕ Addition and Subtraction

Example 1

🧠 The StudyNest BODMAS Method

3️⃣ Multiplication Before Addition

Example 2

4️⃣ Brackets First

Example 3

Why brackets matter

5️⃣ Powers Before Multiplication

Example 4

6️⃣ Roots Within Calculations

Example 5

7️⃣ More Than One Pair of Brackets

Example 6

8️⃣ Brackets Inside Brackets

Example 7

9️⃣ BODMAS with Directed Numbers

Example 8

Example 9

🌍 Real-Life Example: A Shopping Bill

Example 10

🌍 Real-Life Example: Football Points

Example 11

🔎 Comparing Similar Calculations

Without brackets

With brackets

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress