Like everything in our lives, we have certain rules to follow, and make things easy for ourselves. These rules give us the structure, and tell us what and how to use them to get the desired outcomes. In the same way, there are rules in mathematics too that – when followed – help us to get correct answers.
Specifically in algebra – when you are evaluating a problem – we need to follow a specific order in which the various operators must be used. This order of operations is very important, and can be remembered as B O D M A S meaning work out what is in the Brackets ( ) Of Division / Multiplication (whichever operator comes first when read left to right), lastly Addition/Subtraction (whichever operator comes first when read left to right).
A scientific calculator uses this method to solve a given set of operations, and this rule specifies the order in which two or more operators must be performed:
- the expression within the grouping symbols (like brackets, paranthesis, braces) are performed first, then
- multiplication and division are worked out from left to right, and finally
- addition and subtraction are worked out from left to right.
Let’s look at a few examples:
Example 1 : Evaluate 20 + 12 x 3 ÷ 6.
Applying the order of operations rules, this will be 20 + (12 x 3 ÷ 6).
= 20 + (36 ÷ 6)
= 20 + 6
Notice we have first found the value of 12 x 3, and divided the number by 6, and added 20 to the number obtained from this calculation to get the final answer.
We also need to remember that multiplication and division are worked from left to right, i.e. if the multiplication part comes first (i.e. left of division), then we do the multiplication part first before doing the division. Similarly for addition and subtraction.
Example 2 : Evaluate 6 + 4 x 2
Answer: Applying the order of operations, the expression can be represented as 6 + (4 x 2)
So first find (4 x 2) = 8 ; next add 6 to 8; 6 + 8 = 14.
The answer for the expression 6 + 4 x 2 is 14.
Example 3 : Find (2 + 4) ÷ (6 – 3)
Answer: First we work out answers for the two brackets viz. (2 + 4) = 6 and (6 – 3) = 3
So (2 + 4) ÷ (6 – 3) = 6 ÷ 3, i.e.
= = 2.
Example 4 : Evaluate -4 + (3 + 1)
Answer: First work out the bracket (3+1) = 4; then -4 + (3+1) = -4 + 4 = 0.
Example 5 : = ?
Answer: First we find the value of the denominator; 14 – 7 = 7; next = 2.
So = 2
Example 6 : What is ?
Answer: To find the answer, we have to find the value of the numerator (top numbers) and denominator (bottom numbers) separately. Then we divide the numerator value with the denominator value.
So 30 + 10 = 40, and 30 – 10 = 20.
Hence = = 2
We will use a variation of the above order of operations (called B I D M A S) meaning Bracket Index/Indices Division/Multiplication (whichever operator comes first when read left to right) Addition/Subtraction (whichever operator comes first when read left to right). BIDMAS is used when the calculation involves working with indices.
Here are a few examples:
Example 1 : Evaluate 5 x 22
Answer: The BIDMAS order of operations rules say that we have to first work out the indices, i.e. 22 = 4, and then multiply it with 5. So 5 x 4 = 20 is the final answer.
Example 2 : What is 32 + 42
Answer: The first step is to work out the respective indices, i.e. 32 = 9 and 42 = 16; and then add the two numbers. So 32 + 42 = 9 + 16 = 25 is the final answer.