**Introduction**

In simplifying mathematical expressions consisting of the same type of operation, we perform one operation at a time generally starring from the left towards the right. If an expression has more than one fundamental operation, you cannot perform operations in the order they appear. Some operations have to be performed before than the others. This is each operation has its own precedence. Generally, the order in which we perform operations sequentially from left to right is division, multiplication, addition, subtraction. But when the expressions make use of brackets as well, we have a set of rules that defines the precedence of the operations. Let us learn the manner in which this rule is defined.

Let us understand this through an example.

Suppose we want to find the value of 2 + 3 x 5

There are two ways to find the value of the above expression.

In the first method, let us move from left to right, solving the expression in that order. This means, first we will find the sum of 2 and 3. We will get, 2 + 3 = 5.

Now we have 2 + 3 x 5 = 5 x 5

Now, we find the product of 5 with 5 to get the answer as 25. Therefore, we have,

2 + 3 x 5 = 25 ……………………………….. ( 1 )

Now, let us find the value of this expression through another method.

We will first find the value of 3 x 5 and add this product to 2. So, we get,

3 x 5 = 15 and 15 + 2 = 17

Hence, now we have,

2 + 3 x 5 = 17 ………………………… ( 2 )

From ( 1 ) and ( 2 ) we can see that both the results obtained are not the same. This validates the need for having precedence of operators so as to receive the same answer to the given algebraic expression. This precedence is known as the order of operations.

**What is the Basic Order of Operations?**

Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations. Without consistent order of operation, one can make big mistakes during computation. There are a number of rules that define the order of operations, depending upon the involvement of the operators, brackets exponents and other mathematical symbols for operations. This rule is known as DMAS, where

D stands for Division ( ÷ )

M stands for Multiplication ( x )

A stands for Addition ( + )

S stands for subtraction ( – )

Let us understand the rule in further detail.

**What is DMAS?**

MDAS, as defined above is the basic order of operations. In other words, it defines the order in which operations on numbers are to be performed when there are situations where we have two more than two operations in an expression. So, we have the order of operations as –

Let us consider an example. Suppose we want to find the value of 3 + 5 x 2.

Here, we have two operators, namely ( x ) and ( + ). We know that in mathematics, we can perform operations between two numbers only in one go. If we have more than 2 numbers, the result from the first operation is used for performing the next operation. So, in the above example, there are two operations to be performed, ( 3 + 5 ) and ( 5 x 2 ). We need to decide, which one to start with. This is where the MDAS rule comes into play.

From this rule, we can see that Multiplication precedes addition. Therefore, we will perform the operation multiplication first. The result thus obtained will be added to 3. The steps involved in this process will be –

3 + 5 x 2

= 3 + 10

= 13

**Hence, 3 + 5 x 2 = 13**

Let us consider another example.

**Example**

Simplify the following expression using the Order of Operations –

6 + 5 – 4 x 2

**Solution**

We have been given the expression 6 + 5 – 4 x 2

It is important to note that there are three operations involved. Based on the order of operations, multiplication takes precedence over addition and subtraction so we will multiply first. Next, add then subtract since the operation of addition comes before subtraction from left to right.

Therefore, we have,

6 + 5 – 4 x 2

= 6 + 5 – 8 ( Here we have solved 4 x 2 which has given us the result 8 )

= Next, we will add 6 and 3 to get 9. So, we have,

6 + 5 – 8 = 11 – 8 = 3

**Hence, 6 + 5 – 4 x 2 = 3**

**Use of Brackets in Order of Operations**

We have just learnt about the precedence of fundamental operations of addition, subtraction, multiplication and division. According to it the order in which the operations are to be performed is first division then multiplication after which addition and finally subtraction but sometimes in complex operations require a set of operations to be performed prior to the other. For example, if we want the addition to be performed before a division or multiplication then we need to use a bracket.

**Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.** Bracket indicates that the operations within it ought to be performed before the operations outside it. For example, the expression 24 ÷ 3 x 4 would generally be solved as –

24 ÷ 3 x 4

= 8 x 4 = 32

However, If we wish to multiply 3 and 4 first and then divide 24 by the resulting number we write the expression as

24 ÷ ( 3 x 4 )

Now, according to the order of operations, we will solve the brackets first to get,

24 ÷ ( 3 x 4 )

= 24 ÷ 12

= 2

In complex expressions sometimes it is necessary to have (within) in the same time of (one within the another can be confusing for different types of brackets are used most commonly used) are

**Brackets**** ****Name**

( ) Parenthesis of Common Brackets

{ } Braces or Curly Brackets

[ ] Brackets or Square Brackets

It is important to note here that, the left part of each bracket symbol indicates the start of the bracket and the right part indicates the end of the bracket. In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.

**Removal of Brackets**

In order to simplify expressions involving more than one bracket, we will use the steps as explained below –

- See whether the given expression contains vinculum or not. If vinculum is present then perform operations on the red otherwise go to the next step. A vinculum is a horizontal line placed over an expression to show that everything below the line is one group, for example $\overline{2\: x\: 3}$.
- Now, look for the innermost bracket and perform operations within it.
- Remove the innermost bracket by using the following steps –
- If a bracket is preceded by a plus sign, remove it by writing its terms as they are.
- If a bracket is preceded by a minus sign change the positive sign within it to a negative sign and vice-versa.
- If there is no sign between a number and a grouping symbol then it means multiplication.
- If there is a number before some brackets then we multiply the number inside the brackets with the number outside the brackets.

- Look for the next innermost bracket and perform operations in it. Remove the second in a directed by using the rules given in the above steps Continue this process till all the brackets are removed.

Let us understand the above steps using an example.

**Example**

Simplify: 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

**Solution**

We have been given the expression, 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

We will use the order of operations to solve the given expression.

First, we will remove the innermost bracket to get,

37 – [ 5 + { 28 – 12 } ] …………………… [ Removing the innermost bracket ( ) ]

Next, we will remove the curly braces to get

37 – [ 5 + 16 ] …………………………. [ Removing the curly braces ]

Now, we will remove the square brackets to get,

37 – 21 ………………………… [ removing the square brackets ]

Last, we will find the difference between 37 and 21 to get 16.

**Hence simplification of 37 – [ 5 + { 28 – 12 } ] will result in 16.**

**How to Solve Expressions using Order of Operations?**

Now, that we have understood what we mean by order of operations let us learn about the steps involved in solving mathematical expressions using order of operations. The steps involved are –

- Always start by calculating all expressions within parentheses

The first thing is that we should look for removing any grouping symbols in the algebraic expression. This means that at the top of the list, remember to always** **simplify everything inside the grouping symbols. Examples of grouping symbols are parentheses ( ), brackets, and braces { }. For nested grouping symbols, work it out from the inside and out.

- Simplify all the exponents such as square roots, squares, cube, and cube roots

After having removed the parenthesis, we come to the next step of solving all exponential values in the algebraic expression. Exponential expressions which may include root values such as square roots, squares, cube, and cube roots etc. are calculated or evaluated first before performing any of the four fundamental arithmetic operations, namely: addition, subtraction, multiplication, and division.

- Perform the multiplication and the division starting from left to right

Next, multiply and/or divide whichever comes first from left to right before performing addition and subtraction. This tells us that multiplication and division have a higher level of importance than addition and subtraction.

- Finally, do the addition and subtraction similarly, starting from left to right.

Let us understand it using an example.

**Example** Find the value of the expression 4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 )

**Solution** We have been given the expression 4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 ). We need to solve it using MDAS.

Since the given expression contains brackets; we will solve the brackets first. We will get

4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 )

Now, within the Bracket, we will solve the division section first

4 ( 10 + 15 ÷ 5 x 4 – 2 × 2 )

Next, within the bracket itself, we will solve the multiplication to get

4 ( 10 + 3 × 4 – 2 × 2 )

Now, within the bracket, we will solve the addition to get

4 ( 10 + 12 – 4 )

Next, within the bracket, we will add the numbers 10 and 12 and then subtract 4 from the result to get

4 ( 22 – 4 )

Once the bracket is solved, pick up the number from the outside and solve the ‘Of’ section by multiplication:

= 4 × 18

= 72

**Therefore, 4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 )= 72**

**The steps used can be tabulated as**

Brackets Of | 4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 ) |

Division | 4 ( 10 + 15 ÷ 5 x 4 – 2 × 2 ) |

Multiplication | 4 ( 10 + 3 × 4 – 2 × 2 ) |

Addition | 4 ( 10 + 12 – 4 ) |

Subtraction | 4 ( 22 – 4 ) |

Answer | = 4 x 18 |

72 |

**Real-Life Applications of Order of Operations**

We have learnt that in mathematics, the order of operations helps you find the correct value for an expression. However, it is not restricted to problems in mathematics. Order of operations matters in daily life, too. For example, you use a fixed order of activities to be performed for cooking a dish. Similarly, in chemical reactions, a certain order is to be followed to get the desired results. Coming back to mathematics, there are a lot many days to day activities where use the order of operations of MDAS.

Let us understand it through an example.

**Example**

John buys 2 shirts for £8.00 each. He also buys a pair of jeans for £ 20.00 that gets a £ 3.00 discount. How much does he pay in all?

**Solution**

We have been given that John buys 2 shirts for £8.00 each. He also buys a pair of jeans for £ 20.00 that gets a £ 3.00 discount. We need to find how much John pays for the shirts and the jeans. To find this, we must first place all the values in the form of an expression. We will get,

Price of 2 shirts + Price of the pair of jeans –Discount = Total cost paid by John

Hence,

Total cost paid by John = 2 x 8 + 20- – 3

Here we can see that there are three operations involved. Based on the order of operations, multiplication takes precedence over addition and subtraction so we will multiply first. Next, add then subtract since the operation of addition comes before subtraction from left to right. Therefore, we get,

Total cost paid by John = 16 + 20 – 3 = 36 – 3 = £ 33

**Hence, the total cost paid by John for 2 shirts and a pair of jeans = £ 33**

Now we have learnt how to solve expressions involving more than one mathematical operator. But what if there is a use of brackets in the expression. Do we proceed in the same manner or do we make some additions to the rule? Let us find out.

**Key Facts and Summary**

- Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations.
- MDAS is the basic order of operations where, MDAS, where M stands for Multiplication ( x ), D stands for Division ( ÷ ), A stands for Addition ( + ), S stands for subtraction ( – ).
- Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
- The rules of order of operations are –

- Always start by calculating all expressions within parentheses
- Simplify all the exponents such as square roots, squares, cube, and cube roots.
- Perform the multiplication and the division starting from left to right
- Finally, do the addition and subtraction similarly, starting from left to right.

- Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
- In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.