BODMAS – rule
Order of mathematical Operations
The four basic operation in mathematics, as everyone knows, are addition, subtraction, multiplication and division.
These basic operations are used in algebra with numbers or letters or combination of both.
A number multiplied by a variable, a number or variable is called a term.
An expression is combination of such terms.
Example: In 5a + 2b, 5a and 2b are called terms and 5a + 2b is called an expression.
When we calculate a given sum has two numbers and one operator, calculating straight forward as,
15 x 4 = 60, or 25 + 15 = 40.
But, what happens if When we solve a maths problem which has more than one operation as addition, subtraction or multiplication, division then what do we do first.
For example:
(3+ 12 x 5 – 4 = …)?
Which part do we do first? So, there is a set of simple rules for solving mathematical sums.
The order in which all algebraic expressions should be simplified is known as order of operation.
BODMAS
B…is the short form of Bracket.
O…is the short form of Order.
D…is the short form of Division.
M…is the short form of Multiplication.
A…is the short form of Addition.
S…is the short form of Subtraction.
What is ‘BODMAS’?.
The ‘BODMAS’ is an acronym which stands for, order of mathematical operations, as Bracket, Order, Division, Multiplication, Addition and Subtraction.
In some regions, BODMAS is known as ‘PEDMAS’ (Parentheses, Exponent, Division, Multiplication, Addition and Subtraction) is the synonym of BODMAS.
In the UK learning system, is ‘BODMAS.
In the USA learning system, ‘PEMDAS’, some remember (PEDMAS) too.
It is normally called (Parentheses, Exponent, Multiplication, Division, Addition and Subtraction) or ‘PIDMAS'(Parentheses, Index, Division, Multiplication, Addition and Subtraction).
In other places in the world might use ‘BIDMAS’ (Brackets, Index, Division, Multiplication, Addition and Subtraction).
BODMAS
Brackets → Orders → Division → Multiplication → Addition → Subtraction.
PEDMAS
Parentheses → Exponent → Division → Multiplication → Addition → Subtraction.
Brackets (Any part of calculation inside brackets always come first).
Orders(Operations involving powers or square roots).
Division.
Multiplication.
Addition.
Subtraction.
Are BODMAS and PEMDAS are same?
Yes, BODMAS and PEMDAS are same, the acronym terminology may be different but the calculating sequence remains the same.
When we solve the order of operations of an expression, according to BODMAS rule, if an expression have brackets ((), {}, []), the order of brackets to be simplified is ((), {}, []).
We have first solve, the brackets followed by (powers and roots etc.) then division, multiplication, addition and subtraction.
To better understanding of using BODMAS rule, lets understand with examples:
1. Brackets or parentheses are always solve the first in a given expression.
2. If we have both multiplications and division, then solve operations one by one in the order from left to right or All multiplications and divisions solve from left to right.
3. All additions and subtractions solve from left to right.
Example 1: Solve the following arithmetic expression.
5 + 3 x 2
Solved by student 1.
5 + 3 x 2 = 8 x 2 = 16
Solved by student 2.
5 + 3 x 2 = 5 +6 = 11
Above problem solved by two students, student 1 and student 2, but both are solve the problem differently and result is two different answers.
student 1, solve the operation of addition first then multiplication.
student 2, solve the operation of multiplication first then addition.
When we solve the problem in wrong order, then result will be wrong.
Example 2: Solve 2 x ( 4 + 5) = ?
We need to do the operation, or sum, inside the brackets first, 4 + 5, then multiply the answer by 2.
2 x 9
4 + 5 = 9
2 x 9 = 18
Answer is 18.
If we ignored the brackets and did the sum 2 x 4 + 5,
= 8 +5 = 13
we would get 13.
We can observe that, how brackets change the result.
Brackets
Example 3: Solve (5 + 3) x 4
First we solve inside the brackets 5 + 3 = 8,
then 8 x 4 = 32.
Next do anything involving a power or a square root (these are also called orders,) if there is more than one, again calculating from left to right.
Order
Example 4: Solve 23 + 6 = ?
We need to do the power sum first, then add 6.
23 + 6 = ?
23 = 2 x 2 x 2 = 8
8 + 6 = 14
Division and Multiplication
Next part of the equation is to calculate the division and multiplication.
We know that division and multiplication are follow one another, so we go from left to right in the sum and doing each operation in the order in which it appears.
Example 5: Solve 25 ÷ 5 x 6 ÷ 3
Multiplication and Division perform equally, so we start calculate from left to right side.
25 ÷ 5 x 6 ÷ 3
First solve 25 ÷ 5 = 5,
= 5 x 6 ÷ 3
then do multiplication, 5 x 6 = 30
then 30 ÷ 3 = 10.
The answer is 10.
Example 6: Solve 5 x 6 ÷ 3 + 8
= 5 x 6 ÷ 3 + 8
First solve 5 x 6 = 30,
= 30 ÷ 3 + 8
then do division, 30 ÷ 3 = 10,
= 10 + 8
then move to addition, 10 + 8 = 18
The answer is 18.
Addition and Subtraction
The last step of the equation is to calculate, Addition and Subtraction. Again we know that addition and subtraction are follow one another, so we move from left side to right side.
Example 7: Solve 5 + 7 – 4 + 2
= 5 + 7 – 4 + 2
We start from left to right so first solve 5 + 7 = 12,
= 12 – 4 + 2
then 12 – 4 = 8,
= 8 + 2
then 8 + 2 = 10.
The answer is 10.
Example 8: Solve 8 + 12 – 5 + 14
We start from left to right so first solve 8 + 12 = 20,
= 8 + 12 – 5 + 14
= 20 – 5 + 14
= 15 + 14
then 20 – 5 = 15,
then 15 + 14 = 29.
The answer is 29.
Example 9: Solve 5 + 4 x 6
= 5 + 4 x 6
= 5 + 24 (multiplication)
= 5 + 24 (addition)
= 29 (result)
Example 10: Solve (15 + 24) x 3
= (15 + 24) x 3
= 39 x 3 (brackets)
= 39 x 3 (multiplication) = 117 Result
Example 11: Solve53 + 6 ÷ 2 = ?
= 53 + 6 ÷ 2
= 125 + 6 ÷ 2 (power)
= 125 + 6 ÷ 2 (division)
= 125 + 3 (addition)
= 128
= 128 Result
Example 12: Solve2 + 5 x (3 + 4) ÷ 7 – 9
= 2 + 5 x (3 + 4) ÷ 7 – 9
= 2 + 5 x 7 ÷ 7 – 9 (bracket)
= 2 + 35 ÷ 7 – 9 (multiplication)
= 2 + 5 – 9 (division)
= 7 – 9 (addition)
= – 2 (subtraction)
= -2 Result
Example 13: Solve 48 ÷ 12 + (15 – 9) x 4
= 48 ÷ 12 + (15 – 9) x 4
= 48 ÷ 12 + 6 x 4 (bracket)
= 4 + 6 x 4 (division)
= 4 + 24 (multiplication)
= 28
= 28 Result
Example 14: Solve 46 – 2(18 + 12 ÷ 4 x 3 – 3 x 2) + 15
= 46 – 2(18 + 12 ÷ 4 x 3 – 3 x 2) + 15
= 46 – 2(18 + 12 ÷ 4 x 3 – 3 x 2) + 15 (bracket)
= 46 – 2(18 + 3 x 3 – 3 x 2) + 15 (division)
= 46 – 2(18 + 9 – 6) + 15 (multiplication)
= 46 – 2(27 – 6) + 15 (addition)
= 46 – 2(21) + 15 (subtraction)
= 46 – 42 + 15 (multiplication)
= 4 + 15 (subtraction)
= 19 (addition)
= 19 result
Example 15: Solve5 + [(24 – 4) ÷ (23 + 2)] – 4
= 5 + [(24 – 4) ÷ (23 + 2)] – 4
= 5 + [20 ÷ (23 + 2)] – 4 (bracket)
= 5 + [20 ÷ (8 + 2)] – 4 (power)
= 5 + [20 ÷ 10] – 4 (bracket)
= 5 + 2 – 4
= 7 – 4 (addition)
= 3 (subtraction)
= 3 result
Example 16: Solve (46 + 14) ÷ (5 x 3) – 18 + 7
= (46 + 14) ÷ (5 x 3) – 18 + 7
= 60 ÷ 15 – 18 +7 (bracket)
= 4 – 18 + 7 (division)
= – 14 + 7 (subtraction)
= -7 (subtraction)
= -7 result
Example 17: Solve (65 ÷ 13) + 16 x ( 4 + 6) ÷ 5
= (65 ÷ 13) + 16 x ( 4 + 6) ÷ 5
= 5 + 16 x 10 ÷ 5 (bracket)
= 5 + 160 ÷ 5 (multiplication)
= 5 + 32 (division)
= 37 (addition)
= 37 result
Some examples of algebraic expressions look like this.
1. 3 + 2(5 + 6) – 9/32. 5 + 7(4 – 1) x 8/24 – 63. a + 3(5a – 1) + 3{6(2a + 8)}