ORDER OF OPERATIONS

1. Definition

Order of Operations is a set of mathematical rules that tells us the correct sequence for performing different operations in a mathematical expression containing two or more operations.

These rules ensure that everyone solves an expression in exactly the same way and obtains the same answer, regardless of where or when the calculation is performed.

Without a standard order of operations, the same expression could produce different answers, leading to confusion and incorrect calculations.

The two most commonly used systems are PEMDAS and BODMAS. Although they use different words, they represent the same order of solving mathematical expressions.

Example

Consider the expression:

8 + 4 × 3

If addition is performed first:

(8 + 4) × 3 = 12 × 3 = 36

If multiplication is performed first:

8 + (4 × 3) = 8 + 12 = 20

The correct answer is 20 because multiplication is performed before addition according to the Order of Operations.

2. Meaning of PEMDAS

PEMDAS is an acronym that shows the correct order for solving mathematical expressions.

PEMDAS

Letter	Meaning	What You Do
P	Parentheses	Solve everything inside parentheses first.
E	Exponents	Evaluate powers, indices, roots, and exponents.
M	Multiplication	Multiply from left to right.
D	Division	Divide from left to right.
A	Addition	Add from left to right.
S	Subtraction	Subtract from left to right.

3. Meaning of BODMAS

BODMAS is another acronym used in many countries to represent the same order of operations.

BODMAS

Letter	Meaning	What You Do
B	Brackets	Solve everything inside brackets first.
O	Orders	Evaluate powers, indices, roots, and exponents.
D	Division	Divide from left to right.
M	Multiplication	Multiply from left to right.
A	Addition	Add from left to right.
S	Subtraction	Subtract from left to right.

4. PEMDAS and BODMAS Mean the Same Thing

PEMDAS and BODMAS are simply two different names for the same mathematical rule. Both tell us to solve brackets first, then powers or exponents, followed by multiplication and division, and finally addition and subtraction.

The only difference is the terminology used in different countries.

PEMDAS is commonly used in countries such as the United States.
BODMAS is commonly used in countries such as the United Kingdom, Kenya, Australia, India, South Africa, and many other Commonwealth countries.

Regardless of whether you use PEMDAS or BODMAS, the final answer will always be the same when the rules are followed correctly.

Example

Evaluate:

18 − (6 ÷ 2) × 3

Using PEMDAS

Step 1: Solve the expression inside the parentheses first.

6 ÷ 2 = 3

The expression now becomes:

18 − 3 × 3

Step 2: There are no exponents, so move to multiplication and division.

Multiply:

3 × 3 = 9

The expression now becomes:

18 − 9

Step 3: Perform the subtraction.

18 − 9 = 9

Final Answer = 9

Using BODMAS

Step 1: Solve the expression inside the brackets first.

6 ÷ 2 = 3

The expression now becomes:

18 − 3 × 3

Step 2: There are no orders (powers or roots), so perform multiplication.

3 × 3 = 9

The expression now becomes:

18 − 9

Step 3: Perform the subtraction.

18 − 9 = 9

Final Answer = 9

Both PEMDAS and BODMAS give the same final answer because they follow the same order of operations. The only difference is the names used for the steps.

5. Real-Life Applications of the Order of Operations

The Order of Operations is used whenever calculations involve more than one mathematical operation. Following the correct order ensures accurate and consistent results in everyday life.

Calculating the total cost of several items after applying discounts and taxes.
Preparing recipes that involve multiplying ingredient quantities before adding them together.
Engineering and construction calculations involving measurements, areas, and volumes.
Computer programming, where mathematical expressions must be evaluated correctly.
Business and finance when calculating profits, expenses, interest, and taxes.
Science and physics when evaluating mathematical formulas.
Spreadsheet applications such as Microsoft Excel and Google Sheets.
Calculating distances, speeds, and travel times.

Example

A shop sells a notebook for KSh 250. A customer buys 4 notebooks and later pays an additional KSh 100 for delivery.

The total cost is calculated as:

(4 × 250) + 100

First perform the multiplication:

4 × 250 = 1000

Then add the delivery charge:

1000 + 100 = 1100

Total Cost = KSh 1,100

6. Evaluating Numerical Expressions

Evaluating a numerical expression means finding its final value by performing the mathematical operations in the correct order according to the PEMDAS or BODMAS rule.

When solving an expression, never work from left to right unless the order of operations allows it. Always begin with brackets (or parentheses), then evaluate exponents or orders, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

Following the correct order prevents mistakes and ensures everyone obtains the same answer.

Evaluating numerical expressions is used in everyday life when:

Calculating shopping bills involving discounts and taxes.
Finding the total cost of several items with different prices.
Solving engineering and scientific calculations.
Writing computer programs that perform mathematical calculations.
Calculating measurements in construction and architecture.

Example

Evaluate:

8 + 6 × (9 − 5)

Solution

Step 1: Solve the brackets first.

9 − 5 = 4

Expression becomes:

8 + 6 × 4

Step 2: Perform multiplication.

6 × 4 = 24

Expression becomes:

8 + 24

Step 3: Perform addition.

8 + 24 = 32

Final Answer = 32

7. Evaluating Expressions Involving Fractions

Expressions involving fractions are evaluated using the same PEMDAS or BODMAS rule. The only difference is that fractional operations must also follow the correct rules for adding, subtracting, multiplying, and dividing fractions.

When fractions have different denominators, first convert them to equivalent fractions with a common denominator before performing addition or subtraction.

Example

Evaluate:

½ + ¾ × ⅔

Solution

Step 1: Perform multiplication first.

¾ × ⅔ = 6⁄12 = ½

Expression becomes:

½ + ½

Step 2: Add the fractions.

½ + ½ = 1

Final Answer = 1

8. Evaluating Expressions Involving Decimals

Decimal expressions are evaluated using the same order of operations. Brackets, exponents, multiplication, division, addition, and subtraction are performed in exactly the same order as with whole numbers and fractions.

Example

Evaluate:

6.5 + 2 × (4.2 − 1.7)

Solution

Step 1: Solve the brackets.

4.2 − 1.7 = 2.5

Expression becomes:

6.5 + 2 × 2.5

Step 2: Perform multiplication.

2 × 2.5 = 5

Expression becomes:

6.5 + 5

Step 3: Perform addition.

6.5 + 5 = 11.5

Final Answer = 11.5

9. Word Problems Involving PEMDAS/BODMAS

Many real-life mathematical problems require more than one operation. To solve such problems correctly, first convert the information into a mathematical expression and then evaluate it using the PEMDAS or BODMAS rule.

Example

A school buys 8 boxes of exercise books. Each box contains 25 books. The school already has 40 books in storage. How many books does the school have altogether?

Solution

Step 1: Write the mathematical expression.

(8 × 25) + 40

Step 2: Perform multiplication.

8 × 25 = 200

Expression becomes:

200 + 40

Step 3: Perform addition.

200 + 40 = 240

Final Answer = 240 books

7. Worked Examples

Example 1

Evaluate:

8 + 6 × 4

Solution

Step 1: Identify the operations involved.

The expression contains addition and multiplication.

Step 2: Apply PEMDAS.

Perform multiplication before addition.

6 × 4 = 24

Step 3: Rewrite the expression.

8 + 24

Step 4: Add.

8 + 24 = 32

Final Answer: 32

Example 2

Evaluate:

(15 − 7) × 5 + 18 ÷ 3

Solution

Step 1: Solve the brackets.

15 − 7 = 8

The expression becomes:

8 × 5 + 18 ÷ 3

Step 2: Perform multiplication and division from left to right.

8 × 5 = 40

18 ÷ 3 = 6

The expression becomes:

40 + 6

Step 3: Add.

40 + 6 = 46

Final Answer: 46

Example 3

Evaluate:

5² + 18 ÷ (9 − 6)

Solution

Step 1: Solve the brackets.

9 − 6 = 3

The expression becomes:

5² + 18 ÷ 3

Step 2: Evaluate the exponent.

5² = 25

The expression becomes:

25 + 18 ÷ 3

Step 3: Perform the division.

18 ÷ 3 = 6

The expression becomes:

25 + 6

Step 4: Add.

25 + 6 = 31

Final Answer: 31

Example 4

Evaluate:

¾ + ½ × 8

Solution

Step 1: Perform multiplication first.

½ × 8 = 4

The expression becomes:

¾ + 4

Step 2: Convert the whole number to a fraction.

4 = 16⁄4

Step 3: Add the fractions.

¾ + 16⁄4 = 19⁄4

Step 4: Convert the improper fraction to a mixed number.

19 ÷ 4 = 4 remainder 3

Therefore, 19⁄4 = 4¾

Final Answer: 4¾

Example 5

Evaluate:

6.5 + (3.2 × 4) − 7.8

Solution

Step 1: Solve the brackets.

3.2 × 4 = 12.8

The expression becomes:

6.5 + 12.8 − 7.8

Step 2: Perform addition.

6.5 + 12.8 = 19.3

Step 3: Subtract.

19.3 − 7.8 = 11.5

Final Answer: 11.5

Sample Questions

Attempt the following Questions:

(i) Evaluate the following expression using the correct order of operations:
18 + 6 × (12 − 8)

(ii) Evaluate the following expression:
45 ÷ 5 + 7 × 3 − 8

(iii) Evaluate the following expression involving fractions:
⅔ + ¾ × 8

(iv) Evaluate the following expression involving decimals:
12.5 − 3.5 × (4.2 − 1.2)

(v) A school purchases 6 boxes of mathematical calculators. Each box contains 15 calculators. The school gives away 18 calculators to another department and then buys an additional 12 calculators. Using the correct order of operations, calculate the final number of calculators the school has.

Check The Answers Below:

Take Note :

PEMDAS and BODMAS are two different acronyms that describe the same order of operations. The only difference is the words used to represent the steps.
Always solve expressions inside brackets (or parentheses) before carrying out any other operation.
Evaluate orders (powers, indices, roots, and exponents) immediately after brackets.
Multiplication and division have the same level of priority. Perform whichever appears first when reading from left to right.
Addition and subtraction also have the same level of priority. Perform whichever appears first from left to right.
Never perform addition before multiplication unless brackets require you to do so.
Work through an expression one step at a time to reduce mistakes.
Fractions are treated as numbers during evaluation. Apply the order of operations to fraction expressions just as you would with whole numbers.
Decimal numbers follow exactly the same order of operations as whole numbers and fractions.
When an expression contains both fractions and decimals, perform each operation according to the correct order before converting the final answer if necessary.
Always copy the expression carefully before beginning any calculations to avoid changing signs or numbers.
Use brackets whenever writing expressions to make the intended order of operations clear.
Simplify fractions to their lowest terms whenever possible after completing all required operations.
Round decimal answers only if the question specifically instructs you to do so.
Always check your final answer by reviewing each step to ensure the correct order of operations was followed.

ORDER OF OPERATIONS

1. Definition

Example

2. Meaning of PEMDAS

PEMDAS

3. Meaning of BODMAS

BODMAS

4. PEMDAS and BODMAS Mean the Same Thing

Example

5. Real-Life Applications of the Order of Operations

Example

6. Evaluating Numerical Expressions

Example

7. Evaluating Expressions Involving Fractions

Example

8. Evaluating Expressions Involving Decimals

Example

9. Word Problems Involving PEMDAS/BODMAS

Example

7. Worked Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Sample Questions

Check The Answers Below:

Answers:

Take Note :

☕ Support Smart Maths Notes