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# Order of Operations|Definition & Meaning

**Definition**

A **prescribed sequence of rules** that one must follow when solving any given mathematical expression is called the **order of operations**.

In **solving** any **arithmetic problem** involving different **sets of equations** or **arithmetic expressions** involving a**ddition, subtraction, multiplication, and division,** we **need to follow** some **rules** to decide **which operation** to **evaluate first**, as different orders produce different results.

**Building the Concept**

**Suppose** a mathematical expression involves various **arithmetic operations** like **addition, subtraction, division, and multiplication,** along with expressions like **exponents and brackets. **How would you try to solve the problem?

If we try to **randomly evaluate** the **expression** by **opting random order** of **arithmetic operations,** we can **end up** getting the **wrong result**.

Figure 1 – Concept Illustration of Order of Operations

Let us take an example. **Person 1** tries to **solve** the **mathematical expression** by **prioritizing addition over division** and **multiplication,** and **person 2** tries to **solve the same expression** by **prioritizing multiplication over division** and **addition. As shown in Figure 1**, the **result** **of both** persons will **end up** **giving different results**, which does not make any sense.

This is why** the order of operations** is important. It specifies a **standard operating procedure** for **solving a** **mathematical expression** in a specific order **to ensure** the **same result** for the **same mathematical problem**. We have **made** the **convention** in which **each arithmetic operation** is **evaluated** **with** **respect** to its **priority** or order.

**Two Order of Operations Conventions**

There are generally **two conventions** of the **order of operations** for solving mathematical expressions. The first is known as **PEMDAS,** and the second is known as **DMAS**.

**PEMDAS**

Figure 2 – PEMDAS Convention

**PEMDAS** is an **acronym used** for the **order of arithmetic operations** on mathematical operations.** P** is used for **Parenthesis**, **E** is used for **exponents**, and **M** is used for **multiplication**, **D** is used for **division,** **A** is used for **addition**, **S** is used for **subtraction**.

**First Precedence**

**Parenthesis **{}, [], ()

If a mathematical **expression involves** any of the **above brackets,** then the **expression inside** this **bracket** will be **evaluated first**.

**Second Precedence**

**Exponents**, Let **x be any number** and **y be any power. **Then $x^{y}$ is the exponent.** If** the **mathematical expression** **contains** any **term** **related** to **this representation,** it would be **evaluated** on the **second number** after evaluating the first order that brackets.

**Third Precedence**

If the **mathematical expression** contains a **multiply sign,** we evaluate it after evaluating **brackets and exponents**.

**Fourth Precedence**

If a **mathematical expression** **constitutes** any **division operation, **it is performed after dealing with **brackets, exponents, and multiplication.**

**Fifth Precedence**

**Suppose** there is a **mathematical expression** with** brackets, exponent, division, multiplication, and addition operations; **the **addition operation** will be treated lastly precedence-wise or at** fifth precedence**.

**Sixth Precedence**

**Suppose** there is a **mathematical expression** with** brackets, exponent, division, multiplication, addition, and subtraction. **Then the precedence for using subtraction operation will be at **sixth or last.**

**DMAS**

Figure 3 – DMAS Convention

This is **another convention** used for** precedence** or order of using arithmetic operations. This** convention** is **applied** to a **mathematical expression** that **contains only four** arithmetic **operations** namely a**ddition, subtraction, division, and multiplication**. The precedence is described above.

**First Precedence**

**Suppose** there is a **mathematical expression** **involving** only **division and multiplication. **Then, the **first precedence** will be **given** to the **division operation**.

**Second Precedence**

**Multiplication** has the **second precedence** when treating such problems. If there is an expression involving **multiplication** and **subtraction only,** then **multiplication** must be performed **first**.

**Third Precedence**

**The addition** has the **third precedence** while solving such kinds of problems, but if there is **only an addition** and **subtraction** operation in the expression, then** addition** will have the** first precedence**.

**Fourth Precedence**

**While solving** such** problems described** above, we** give** the **least precedence** to **subtraction** which is **fourth**. In any convention, whether **PEMDAS or DMAS,** **subtraction** precedence **will be last** in any case.

**General SOPs for Order Precedence**

Figure 4 – Sops for Order Precedence for Arithmetic Operations

**SOP 1**

Always **solve brackets first** in any mathematical expression. There are **different types** of **brackets: **round “(),” curly “{},” and square “[].” () are solved **first**, followed by {}, followed by [].

**SOP 2**

Always give the **second precedence** to the **exponents.** For example, if this is the expression (2+3) x 5$\mathsf{^2}$, then **always** solve the **exponent secondly**.

**SOP 3**

Talking about** arithmetic operations** left behind **Multiplication, Addition, Division, and Subtraction. **Then, we will use **multiplication or division** on the **third precedence** after brackets and exponents.

**SOP 4**

**After** solving **brackets, exponents, multiplication, and division, **we **solve addition** and **subtraction** at the end,** respectively**.

**Riddles To Memorize the Order of Operations**

**PEMDAS:****P**arty**E**very**M**onth**D**ance**A**lways**S**unday**DMAS:****D**ance**M**adly**A**fter**S**unday

**Example Problems Involving Multiple Operations**

**Example 1**

a = [(5 + 10) + 4$\mathsf{^2}$ x 4 – 42 + 50] $\div$ 10

**Solve** the above example using the **PEMDAS** convention for **order precedence**.

**Solution**

**First Precedence: **Solve all expressions within **round brackets** (). This is 5 + 10 = **15**.

**Second Precedence: **Solve all expressions within the square brackets. This is 15 + 4$\mathsf{^2}$ x 4 – 42 + 50.

**Third Precedence: **Solve exponent 4$\mathsf{^2}$ = **16**.

**Fourth Precedence: **Multiply 16 by 4 to get **64**.

**Fifth Precedence: **Add 15, 64, and 50 to get **129**, then subtract** 42** to get **87**. This is the result of the expression within [].

**Last Precedence: **Solve expression outside the bracket**. **This is a division by 10, so we will divide **87** with 10 to get **8.7**.

Following the PEMDAS order of operations:

a = [(5 + 10) + 4$\mathsf{^2}$ x 4 – 42 + 50] $\div$ 10

a = [(15) + 4$\mathsf{^2}$ x 4 – 42 + 50] $\div$ 10

a = [15 + 16 x 4 – 42 + 50] $\div$ 10

a = [15 + 64 – 42 + 50] $\div$ 10

a = [129 – 42] $\div$ 10

a = [87] $\div$ 10

a = 8.7

**Example 2**

**Solve** the following **mathematical expressions** using** DMAS**.

a = 28 $\div$ 4 – 15 x 20 + 30

b = 40 – 30 + 12 x 4 + 5 $\div$ 3

### Solution

**Expression A**

**First Precedence: **Divide 28 by 4.

**Second Precedence: **Multiply 15 by 20.

**Third Precedence: **Add 30 with the result of a division of 28 by 4.

**Fourth Precedence: **Subtract 300 from the final result.

Following this order of operations:

a = 28 $\div$ 4 – 15 x 20 + 30

a = 7 – 15 x 20 + 30

a = 7 – 300 + 30

a = -300 + 37

**a = -263**

**Expression B**

**First Precedence: **Divide 5 by 3.

**Second Precedence: **Multiply 12 by 4.

**Third Precedence: **Add 40 with the result of a division of 5 by 4 and multiplication of 12 by 4.

**Fourth Precedence: **Subtract 30 from the final result.

Following this order of operations, we get the following result:

b = 40 – 30 + 12 x 4 + 5 $\div$ 3

b = 40 – 30 + 12 x 4 + 1.666

b = 40 – 30 + 48 + 1.666

b = 89.666 – 30

**b = 59.666**

*All mathematical drawings and images were created with GeoGebra.*