Order of Operations (PEMDAS/BODMAS) - Sixth Grade Mathematics
The order of operations is a universal convention that establishes an unambiguous sequence for evaluating mathematical expressions containing multiple operations.
The Hierarchy of Operations
Without a standardized agreement on which operations to calculate first, an expression like 8 + 4 * 3 could yield two completely different results: 36 (if addition is done first) or 20 (if multiplication is done first). The global mathematical community resolved this ambiguity through the order of operations hierarchy, commonly remembered by the acronyms PEMDAS or BODMAS.
+-------------------------------------------------------------------------+
| THE ORDER OF OPERATIONS HIERARCHY |
+-------+-------------------------+---------------------------------------+
| Level | Operation Group | Priority Rule |
+-------+-------------------------+---------------------------------------+
| 1 | Parentheses / Brackets | Innermost to outermost symbols |
| | ( ), [ ], { }, bars | (Round, Square, Curly, Fraction bars) |
+-------+-------------------------+---------------------------------------+
| 2 | Exponents / Orders | Evaluate all powers and roots |
| | x^2, 10^3, etc. | from left to right |
+-------+-------------------------+---------------------------------------+
| 3 | Multiplication & | Equal priority! Perform strictly |
| | Division | from left to right as encountered |
+-------+-------------------------+---------------------------------------+
| 4 | Addition & | Equal priority! Perform strictly |
| | Subtraction | from left to right as encountered |
+-------+-------------------------+---------------------------------------+
The Acronyms: PEMDAS and BODMAS
PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction. Both acronyms describe the identical mathematical hierarchy.
P / B --> Parentheses / Brackets
E / O --> Exponents / Orders
M & D --> Multiplication & Division (Strictly Left-to-Right!)
A & S --> Addition & Subtraction (Strictly Left-to-Right!)
The Left-to-Right Crucial Rule
The most common misconception in mathematics is believing that multiplication must always precede division, or that addition must always precede subtraction. In truth, multiplication and division are co-equal in rank. Whichever appears first when reading from left to right must be computed first. The same equal-rank rule governs addition and subtraction.
Example of Equal Rank: 24 / 4 * 2
Incorrect interpretation: Doing multiplication first gives 24 / 8 = 3. (WRONG)
Correct interpretation: Moving left to right, division comes first!
24 / 4 = 6
6 * 2 = 12 (CORRECT)
Nested Grouping Symbols and Fraction Bars
When mathematical problems involve complex layers, grouping symbols indicate which components must be resolved before other operations may interact with them.
Layered Hierarchy: { [ ( Innermost ) ] Outermost }
Step 1: Evaluate innermost parentheses: ( ... )
Step 2: Evaluate surrounding brackets: [ ... ]
Step 3: Evaluate outermost braces: { ... }
Working from Inside Out
When an expression contains nested grouping symbols, always identify the innermost pair of parentheses first. Perform all operations within that innermost pair using the standard order of operations. Once that innermost value is a single number, eliminate those parentheses and move outward to the enclosing square brackets or curly braces.
Fraction Bars as Implicit Grouping Symbols
A horizontal fraction bar acts as a grouping symbol for both its entire numerator and its entire denominator. An expression such as (30 + 10) / (7 - 2) written across a fraction bar requires that you evaluate the complete numerator (30 + 10 = 40) and the complete denominator (7 - 2 = 5) before performing the final division (40 / 5 = 8).
Comprehensive Step-by-Step Evaluations
Systematic presentation of calculations line-by-line prevents computational oversights and allows effortless error tracing.
Evaluate: 5 + 2 * (14 - 3^2) + 18 / 3
Step 1: Focus inside parentheses: (14 - 3^2)
Evaluate exponent first: 3^2 = 9
Then subtract: 14 - 9 = 5
Expression becomes: 5 + 2 * 5 + 18 / 3
Step 2: Perform Multiplication and Division from left to right:
First multiply: 2 * 5 = 10
Next divide: 18 / 3 = 6
Expression becomes: 5 + 10 + 6
Step 3: Perform Addition from left to right:
5 + 10 = 15
15 + 6 = 21 (Final Value)
Exercises
Exercise 1. Evaluate the numerical expression 42 - 12 / 3 + 4 * 2. Explain the order of steps.
Exercise 2. Evaluate the expression with nested brackets: 5 * [32 - (4^2 - 7)] / 3.
Exercise 3. Insert parentheses into the expression 6 + 4 * 5 - 2 to make its value equal to 48. Show the verification of your placement.
Solutions
Solution to Exercise 1. First, scan for grouping symbols and exponents; there are none. Next, evaluate multiplication and division from left to right. The first operation encountered is 12 / 3 = 4. The next is 4 * 2 = 8. Substitute these back to get 42 - 4 + 8. Finally, perform addition and subtraction from left to right: first subtract 42 - 4 = 38, then add 38 + 8 = 46. The final answer is 46.
Solution to Exercise 2. Begin with the innermost parentheses: (4^2 - 7). Evaluate the exponent: 4^2 = 16. Then subtract: 16 - 7 = 9. The expression simplifies to 5 * [32 - 9] / 3. Next, evaluate inside the square brackets: 32 - 9 = 23. The expression becomes 5 * 23 / 3. Now compute multiplication and division from left to right: first multiply 5 * 23 = 115. Finally, divide 115 by 3 to obtain 115/3, which is 38 and 1/3 (or approximately 38.33).
Solution to Exercise 3. Without parentheses, 6 + 4 * 5 - 2 evaluates to 6 + 20 - 2 = 24. To reach 48, place parentheses around (6 + 4) and around (5 - 2) or test other groupings. Notice that 48 can be factored into 8 * 6 or 16 * 3. Testing (6 + 4 * 5) - 2 = 26 - 2 = 24 does not work. Testing (6 + 4) * (5 - 2) gives 10 * 3 = 30. Testing [ (6 + 4) * 5 ] - 2 gives 50 - 2 = 48! Inserting brackets and parentheses as ((6 + 4) * 5) - 2 produces (10 * 5) - 2 = 50 - 2 = 48. Alternatively, grouping as (6 + 4 * 5) - 2 = 24; but writing (6 + 2) is not permitted. Therefore, ((6 + 4) * 5) - 2 = 48 is the verified correct placement.