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# PEMDAS

## 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 another rule that defined the precedence of the operations which is known as PEMDAS. Let us learn the manner in which this rule is defined.

## What is PEMDAS?

PEMDAS is a mnemonic device used to remember the correct order of mathematical operations to be used if there are multiple operations to be calculated in an expression. PEMDAS stands for Parentheses, Exponents, Multiplication or Division, and Addition or Subtraction. Thus, if an expression has parentheses or any grouping symbol, this means we need to prioritize the operations inside it before the remaining operations.

The PEMDAS Rule is a set of rules that prioritize the order of calculations, that is, which operation to perform first. Otherwise, it is possible to get multiple or different answers. 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.

### The Importance of PEMDAS

Without PEMDAS, there are no guidelines to obtain only one correct answer. We have seen in the above example that without the PEMDAS rule, a different expression would result in different values when performed in a different order. Hence, the PEMDAS rule has been defined for solving mathematical expressions.

## The Ambiguous Case

To be clear, we can see from the table that Parentheses is not a mathematical operation but rather a grouping symbol, just like the Brackets ([]) and Braces ({}). Thus, we must not treat them as multiplication symbols because the expression could have more than one answer and will lead to ambiguity and bewilderment. But as we all know, basic Math problems such as operations must give one answer.

The best examples are the viral math problems that may also scratch your head so many times. Just like this one:

8÷2(2+2)

If we follow the traditional method, this will be the order of operation:

1

With this method, we have 8÷2(2+2)=1.

How about if we follow the “modern” definition of PEMDAS, treating Parentheses as a multiplication symbol still?

Following this manner, we have 8÷2(2+2)=16.

Now, you may wonder why we got different answers and what the real answer is. Well, the answers lie with the basic principle of using grouping symbols such as Parentheses. First, let us assume we have numbers that will be named as x and y. To multiply them, we have

x ×y

But x is almost identical to that could lead to confusion when written. Thus, the multiplication dot ( ∙ ) is being used for multiplication. Moreover, attaching the terms to multiply them (e.g. x and y) are commonly used just like below: xy

Using this notation of multiplying of expressions, another question will arise: How about if there is one group of expressions being multiplied to another group of terms? That is when grouping symbols come in. Hence, the way we see the groups of binomials that use FOIL METHOD to be solved.

We cannot attach y to x in the middle because it will signify that y must be multiplied by x.

x+yx-y

Of course, this will give us an incorrect interpretation of what should be done for this expression to be solved.

Therefore, the use of parentheses to group the expressions to be multiplied (which is correct) led also to the erroneous interpretation that a single term or number inside the parentheses must be multiplied by default to the numbers before or after it.

Additionally, this is the reason why some calculators perceive parentheses as a multiplication symbol by default. In the calculator’s perspective, any term before or after the parenthesis is another group to be multiplied or also have a parenthesis, even though we did not type it.

Case 1: 8÷2(2+2)

What we typed: 8÷2(2+2)

Some calculator’s mind: 8÷(2)(2+2)

Case 2: 8÷2×(2+2)

What we typed: 8÷2×(2+2)

Some calculator’s mind: 8÷(2)x(2+2)

Case 3: 8÷2(2+2)

What we typed: 8÷2(2+2)

Modern calculator’s mind: (8÷2)(2+2) Treating it as

(4)(4)

16

As a result, some are now proposing the use of either the times sign () or the multiplication dot () to straighten out the confusion if parenthesis is present. For instance, if you are searching for the definition of FOIL Method, you may have seen the illustration below on Wikipedia*.

Going back to the expression 8÷2(2+2), if the person who made the problem implies that 2 alone must be multiplied by the group of numbers inside the parenthesis ((2+2)), the correct expression must be like the ones below.

8÷2×(2+2)   or   8÷2∙(2+2)

Thus, we can also translate the part of the expressions with mathematical symbols attached with:

+(2+2)           (2+2) is being added

– (2+2)          (2+2) is being subtracted from

x (2+2)          (2+2) is being multiplied by

÷ (2+2)          (2+2) is being divided by

### 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.

One of such rules is PEMDAS, where

P stands for Parentheses

E stands for Exponents

M stands for Multiplication

D stands for Division

A stands for Addition

S stands for Subtraction

## Rules of PEMDAS

The following rules should be followed while finding the value of any algebraic expressions using the PEMDAS rule –

1. 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.

1. 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.

1. 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.

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

The “Left to Right” Order of Operations

According to the rule of PEMDAS, if Multiplication and Division or Addition and Subtraction are consecutive or next to each other in the expression, we must operate them from the left to right. Let us use the expression 8÷2×(2+2) to know the reasons behind the rule.

8÷2×(2+2) could also be written as  $\frac{8}{2}$ x (2+2). This implies that we should divide 8 by 2 first and multiply the quotient by the product inside the parentheses. Also, unlike Multiplication, Division is not commutative, which means that we do need to prioritize it if written first.

Left-to-right

8÷2×(2+2)

8÷2×4

8÷24

44

16 Right-to-left

8÷2×(2+2)

8÷2×4

8÷2×4

8÷8

1 How about if we have multiplication written first?

8×2÷(2+2)

Left-to-right

8×2÷(2+2)

8×2÷4

8×24

164 Right-to-left

8×2÷(2+2)

8×2÷4

8×2÷4

8×12 From the comparisons above, if the Division goes before the Multiplication, we must operate them from left to right to obtain an accurate answer. However, if Multiplication precedes before Division, either way, we could still get the correct answer. However, for the sake of simplicity (just like you may have noticed from the fourth method), operating from left to right will make the calculations easier.

Now, let us explore if this is the same reason for Addition and Subtraction. Let us use the expression 12+3-2.

Left-to-right

12+3-2

15-2

13 Right-to-left

12+3-2

12+1

13 13 Z

How about if we have subtraction written first?

12-3+2

Left-to-right 12-3+2

9+2

11 Right-to-left

12-3+2

12+5 Again, we got the same answers either operating from left to right or vice versa if Addition goes first. But the case was different if Subtraction precedes before Addition. For this reason, if these operations are consecutive in an expression, we should operate them from left to right.

How about if we combined them all? Let us see if this will make some difference.

Example: 7×8+6-5

Left-to-right

7×8+6-5

56+6-5

56+6-5

62-5

57 Right-to-left

7×8+6-5

7×8+1

7×8+1

7×9

63 Following the MDAS, then the Right-to-left Rule

7×8+6-5

56+6-5

56+6-5

56+1

57 From the comparisons above, operating literally from right-to-left if we have all the operations in an expression will lead to an incorrect answer. Thus, we should prioritize Multiplication or Division before Addition or Subtraction if we have them all in an expression. This rule is a must if the operations are mixed up. Let us have an example.

Example: 7×8+6-5÷5

Following the MDAS

7×8+6-5÷5

56+6-1

56+6-1

62-1

61 Left-to-Right

7×8+6-5÷5

56+6-5÷5

56+6-5÷5

62-5÷5

62-5÷5

57÷5

$\frac{57}{5}$ In a nutshell, we need to follow the left-to-right rule because of the precedence of Multiplication or Division and Addition or Subtraction in an expression. Moreover, we need to put a mathematical symbol before or after a grouping symbol such as Parentheses (()), Brackets ([]), and Braces ({}) if the terms around them are not grouped terms.

Let us understand the above rules using an example.

Example

Simplify the following expression using the Order of Operations PEMDAS –

6 + 3 – 4 x 2

Solution

We have been given the expression 6 + 3 – 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, subtract then add since the operation of subtraction comes before addition from left to right.

Therefore, we have,

6 + 3 – 4 x 2

= 6 + 3 – 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 + 3 – 8 = 9 – 8  = 1

Hence, 6 + 3 – 4 x 2 = 1

We have now learnt what we mean by PEMDAS. It is important to note here that the first alphabet of the PEMDAS rule, i.e. Parenthesis holds a significant importance in solving algebraic expressions. There are certain rules that should be taken care of when removing these parentheses or brackets are they are commonly known. Let us now learn about these rules.

## Use of Brackets

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 PEMDAS rule, 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

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.

Let us now understand the rules that need to be followed while removing brackets in an algebraic expression

### Removal of Brackets

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

1. 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 $\overline{2 x 3}$.
2. Now, look for the innermost bracket and perform operations within it.
3. Remove the innermost bracket by using the following steps –
1. If a bracket is preceded by a plus sign, remove it by writing its terms as they are.
2. If a bracket is preceded by a minus sign change the positive sign within it to a negative sign and vice-versa.
3. If there is no sign between a number and a grouping symbol then it means multiplication.
4. If there is a number before some brackets then we multiply the number inside the brackets with the number outside the brackets.
4. 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 PEMDAS 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.

## Calculating Expressions with Exponents

E in PEMDAS stands for Exponents. In most cases, we could see exponents inside the Parentheses or any other grouping symbols. Primarily, Exponents are short notations from multiplying the same number twice or more.

3×3×3×3

34

So you may ask, “Why it must be done before doing Multiplication and Division?” Let us have an example to see why.

81÷34

With this example, it is clear that we cannot divide 81 by 34 right away because we do not know yet what is the exact value of 34. Otherwise, if we break down 34 in the expression, this will give us

81÷34

81÷(3×3×3×3)

At this point, you may notice that we grouped inside the Parentheses the terms we split. It is because, technically, these numbers are from one single term, i.e. 34

81÷3×3×3×3

27×3×3×3 81÷(3×3×3×3)

81÷81 And since the terms are now inside the parentheses, keep in mind that in the rule of PEMDAS, we must solve first the value inside it before the other operations that are present in an expression.

From what we have seen, it is important to solve first the terms with exponents to obtain their exact value and simplify the process of calculating the expression.

## Negative Integers inside the Parentheses

You may notice that if we finished solving the operations inside the parentheses, we remove the parentheses and drop alone the single term calculated. The basis is by definition, we only use Parentheses for a group of terms or expressions. So, if the terms inside were squeezed out, it will turn to a single term (not a group anymore).

What if we got negative terms being simplified? Keep in mind that technically, any negative term is multiplied by -1 by default. So, it is still right to use a convention of  -1 especially if we have a plus (+) or minus (-) sign before it. Furthermore, if you are already studying Algebra, you may know that any algebraic term has a coefficient of 1. Since parentheses are perceived to be multiplied essentially, we need to follow the formula of multiplying integers.

+×- = –

-×- = +

Example 1: 12÷3+(5-7)

12÷3+(5-7)                  Solve the expression inside the parentheses first

12÷3+(-2)

12÷3+(-2)                       Operate division next

4+1(-2)                               Multiply (-2) by  +1

4-2

2

Example 2: 9+5-(-10+5)

9+5-(-10+5)                Solve the expression inside the parentheses first

9+5-(-5)

9+5-1(-5)                          Multiply (-2) by  -1

9+5+5

19

## Various Examples Using the PEMDAS Rule

Now that everything is clear, let us explore various examples of expressions to practice the PEMDAS rule the right way.

Example 1: 5×6-2×8+4

5×6-2×8+4                Solve the expressions inside the parentheses first

30-212

30-2×7                                   Operate multiplication next

30-24

6

Example 2: 24÷(7-3)+5

24÷(7-3)+5                         Solve the expression inside the parentheses first

24÷4+5

24÷4+5                                     Simplify the term with exponent

16÷4+5

16÷4+5                                     Operate division next

4+5

9

Example 3: 10+(32-6)×7

10+(32-6)×7                        Solve the expression inside the parentheses first

10+(9-6)×7                          Simplify the term with exponent

10+3×7

10+3×7                                      Operate multiplication next

10+21

31

Example 4: $(\frac{1}{2}+\frac{2}{3}) x \frac{1}{7}$

$(\frac{1}{2}+\frac{2}{3}) x \frac{1}{7}$     Solve the expression inside the parentheses first

$(\frac{3}{6}+\frac{4}{6}) x \frac{1}{7}$     Find the common denominator

$\frac{7}{6} x \frac{1}{7}$

$\frac{1}{6}$

Example 5: $2÷\frac{3}{4}-(\frac{5}{8}-\frac{3}{8})$

$2÷\frac{3}{4}-(\frac{5}{8}-\frac{3}{8})$     Solve the expression inside the parentheses first

$2÷\frac{3}{4}-\frac{2}{8}$

$2÷\frac{3}{4}-\frac{2}{8}$                                         Operate division of fractions next

$\frac{1}{2} x \frac{3}{4}-\frac{2}{8}$

$\frac{3}{8}-\frac{2}{8}$

$\frac{1}{8}$

Example 6: (33-4) x (26÷13)+(9-5)

(33-4) x (26÷13)+(9-5)         Solve the expressions inside the parentheses first

(9-4) x 2+4                                       Simplify the term with exponent

5×2+4

5×2+4                                                   Operate multiplication next

10+4

14

Example 7: -50+60÷(52-15)

-50+60÷(52-15)                           Solve the expression inside the parentheses first

-50+60÷(25-15)                           Simplify the term with exponent

-50+60÷10

-50+60÷10                                          Operate division next

-50+6

-44

Applications of the PEMDAS Rule

There are many ways where this rule can be applied. As a matter of fact, there is a chance that you are applying it without realizing it. But the common denominator of these situations is we need to apply the correct order of multiple operations to obtain an accurate result.

Situation 1: Grocery Shopping

Maya added to her cart the following: 6 packs of ramen noodles (\$4 each), 4 cans of mushroom soup (\$1.5 each), and 2 loaves of bread (\$2.5 each). She wants to know what should be typed in her mobile calculator to know what will be her change if she paid$50. What will be the correct expression Maya should type to her calculator?

Solution:

Ramen noodles         6× (\$4) Mushroom soup 4× (\$1.5)

Loaf of bread             2× (\$2.5) We need to add all of the expressions above to get the sum. Since the sum of all expressions will be subtracted from the$50 bill, we need to group or enclose them with a parenthesis.

6×4+4×1.5+2×2.5                 To sum up the price of all items

50-(6×4+4×1.5+2×2.5              To subtract the sum from the amount of bill )

Hence, Maya should type in 50-(6×4+4×1.5+2×2.5.)

Situation 2: Solving an Algebraic Equation

If you are currently learning Algebra, perhaps you encounter a lot of problems where the PEMDAS rule is needed to solve an equation. A common example is the one below.

18÷6×b-4=17

18÷6×b-4=17                                   MD is present, so operate from left to right

3×b-4=17

3×b-4=17

3b-4=17

3b-4+4=17+4                            Add 44 to both sides

3b=21

$\frac{3b}{3}=\frac{21}{3}$                                    Divide both sides by 3

b=7

Situation 3: Conversion of Units of Temperature

Scientific formulas such as Celsius to Fahrenheit and vice versa use the PEMDAS rule as well. For instance, if someone visited the United States from another country that uses Celsius typically may wonder what the temperature is that day in Fahrenheit (which the U.S. normally uses). For instance, her smartphone says that it is currently 25℃ in her area, what is its value in Fahrenheit?

Formula to convert Celsius to Fahrenheit:

℉=℃∙$\frac{9}{5}$+32

Solution 1:

℉=25°∙$\frac{9}{5}$+32

℉=25°∙$\frac{9}{5}$+32                     Multiply the whole number by the numerator of the following term

℉=$\frac{225°}{5}$+32

℉=$\frac{225°}{5}$+32                          Operate division next

℉=45°+32

℉=77°

Solution 2:

℉=25°∙$\frac{9}{5}$+32

℉=25°$\frac{9}{5}$+32 Divide the whole number by the denominator of the following term

℉=5°∙9+32

℉=5°∙9+32                       Operate multiplication next

℉=45°+32

℉=77°

Activity Level 1

(This activity is suitable for Grades 1-6)

Directions: Calculate the following expressions.

Solutions for Activity Level 1:

1. 20÷4+(3×5)                 Solve the expression inside the parentheses first

20÷4+15

20÷4+15                          Operate division next

5+15

20

2. (62-12)÷2-7                Solve the expression inside the parentheses first

(36-12)÷2-7                Simplify the term with exponent

24÷2-7

24÷2-7                              Operate division next

12-7

5

3. 7×5-14÷2                      Operate multiplication and division first

35-7

28

4. 100-(150÷3)÷52          Solve the expression inside the parentheses first

100-50÷52                         Simplify the term with exponent

100-50÷25

100-50÷25                         Operate division next

100-2

98

5. (40÷8) x (32+1)                Solve the expressions inside the parentheses first

5 x (32+1)                              Simplify the term with exponent

5 x (9+1)                                Solve the remaining expression inside the parentheses

5 x 10

50

6. 6+43÷16                              Simplify the term with exponent first

6+64÷16

6+64÷16                              Operate division next

6+4

10

7. $\frac{1}{2} x \frac{1}{4}$÷3                                     Apply multiplication first

$\frac{1}{8}$÷3

$\frac{1}{8} x \frac{1}{3}$

$\frac{1}{24}$

8. ($\frac{5}{2}+\frac{3}{2}$)÷2×8                             Solve the expression inside the parentheses first

4÷2×8

4÷2×8                                       MD is present so operate from left to right

2×8

16

9. 72+($\frac{3}{2} x \frac{2}{3}$)×2                         Solve the expression inside the parentheses first

72+1×2                                    Simplify the term with exponent

49+1×2

49+1×2                                    Operate multiplication next

49+2

51

10. (800-350)-(2×45)+(72÷23)        Solve the expressions inside the parentheses first

450-90+(72÷8)                                      Simplify the term with exponent

450-90+9

450-90+9                                                    AS is present, so operate from left to right

360+9

369

Activity Level 2

(This activity is suitable for Grades 7-12)

Directions: For numbers 1-5, calculate the following expressions. Then, solve the problems for numbers 6-7.

1. 3+60÷(8-11)
1. -5+(53÷25)×4
1. ((16÷5+3)×2)÷4
1.  -7-9+(54÷18)×6
1. (2+$\frac{4}{5}$) x ($\frac{1}{7} x \frac{5}{2}$)×-3
1. From the equation 13×3-2×a+5=60, what is the value of a?
1. On average, the neighboring areas of the United States can reach 68℉. What is its value in Celsius?

Solutions for Activity Level 2:

1. 3+60÷(8-11)                      Solve the expression inside the parentheses first

3+60÷(-3)

3+60÷(-3)                             Operate division next

3-20

17

1. -5+(53÷25)×4                    Solve the expression inside the parentheses first

-5+(53÷25)×4                    Simplify the term with exponent

-5+(125÷25)×4                 Calculate the expression inside the parentheses

-5+5×4

-5+5×4                                    Operate the multiplication next

-5+20

15

1. ((16-5+3)×2)÷4              Solve the expression inside the parentheses first

((16-5+3)×2)÷4              AS is present, so calculate from left to right

((11+3)×2)÷4

((11+3)×2)÷4

(14×2)÷4                                 Simplify the expression inside the parentheses

28÷4

7

1.  -7-9+(54÷18)×6            Solve the expression inside the parentheses first

-7-9+3×6

-7-9+3×6                             Operate multiplication next

-7-9+18

-7-9+18                                  AS is present, so calculate from left to right

-16+18

1. (2+$\frac{4}{5}$) x ($\frac{1}{7} x \frac{5}{2}$)×-36            Solve the expressions inside the parentheses first

($\frac{10}{5}+\frac{4}{5}) x (\frac{5}{14}$)×-3                   Find the common denominator

($\frac{14}{5} x \frac{5}{14}$)×-3

($\frac{14}{5} x \frac{5}{14}$)×-3                                   Operate multiplication next

1×-3

-3

1. From the equation 13×3-2×a+5=60, what is the value of a?

13×3-2×a+5=60

13×3-2×a+5=60                           Apply multiplication first

39-2a+5=60

39-2a+5=60                            Add the similar terms

-2a+44=60

-2a+44-44=60-44                 Subtract 44 from both sides

-2a=16

$\frac{-2a}{-2}=\frac{16}{-2}$                            Divide both sides by -2

a=-8

1. As of today, the neighboring areas of the United States have the temperature of 68℉. What is its value in Celsius?

The formula for Fahrenheit to Celsius:
℃=$\frac{5}{9}$.(℉-32°)

Method 1:

℃=$\frac{5}{9}$ . (68°-32°)                Solve the expression inside the parentheses

℃=$\frac{5}{9}$ . 36°

℃=$\frac{5}{9}$∙36°                                Multiply the preceding numerator by the whole number

℃=$\frac{180°}{9}$

℃=20°

Method 2:

℃=$\frac{5}{9}$. (68°-32°)                 Solve the expression inside the parentheses

℃=$\frac{5}{9}$ . 36°

℃=$\frac{5}{9}$ .36°                                 Divide the whole number by the preceding denominator

℃=5∙4°

℃=20°

Therefore, the neighboring areas of the United States have an average temperature of 20℃ as of today.

## Solved Examples

Example 1 Simplify: 197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ]

Solution We have been given the algebraic expression,

197 – [1/9 { 42 + (56 –  $\overline{8+9}$ ) } +108 ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

197 – [1/9 { 42 + (56 – 17 ) } +108 ]

Next, we will remove the innermost bracket to get,

197 – [1/9 { 42 + 39 } +108 ]

Now, we will remove the curly braces to get

197 – [81 / 9  +108 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

197 – [9  + 108 ]

Next, we will remove the square brackets to get,

197 – 117

Last, we just need to find the difference of the remaining two values to get,

80.

Hence, simplification of 197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ] = 80.

Example 2 Simplify 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

Solution We have been given the expression

95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 –  $\overline{17-10}$ } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

95 – [ 144 ÷ ( 12 x 12 ) – ( – 4 ) – { 3 – 7 } ]

Next, we will remove the innermost bracket to get,

95 – [ 144 ÷ 144 + 4  – { 3 – 7 } ]

Now, we will remove the curly braces to get

95 – [ 144 ÷ 144 + 4  + 4 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

95 – [ 1 + 4  + 4 ]

Next, we will remove the square brackets to get,

95 – 9

= 86

Hence, simplification of 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ] = 86

Example 3 Simplify ( 1 + 3 ) 2 – 2 ( 4 – 7 )

Solution We have been given the expression

( 1 + 3 ) 2 – 2 ( 4 – 7 )

We can see that there is a parenthesis in the expression as well as an exponent. So, going by the PEMDAS rule, we will first solve the parenthesis to get,

( 4 ) 2  + 6

Next, we will solve the exponent in the expression to get,

16 + 6

Last, we are left with the addition operation after which we will get the result as 2.

Hence, simplification of ( 1 + 3 ) 2 – 2 ( 4 – 7 ) = 22

Example 4 Simplify 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Solution We have been given the expression

15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

15 – ( – 5) { 4 – 4 } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Next, we will remove the innermost bracket to get,

15 + 5 x 0  ÷ [ 3 { 5 + 18 } ]

Now, we will remove the curly braces to get

15 + 0  ÷ [ 3  x 23 ]

Next, we will remove the square brackets to get,

15 + 0  ÷ 69

Now, it is important to see that we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

15 + 0

= 15

Hence, simplification of 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ] = 15

## Key Facts and Summary

1. The PEMDAS Rule is a set of rules that prioritize the order of calculations, that is, which operation to perform first.
2. PEMDAS is an acronym where P stands for Parentheses, E stands for Exponents, M stands for Multiplication, D stands for Division, A stands for Addition and S stands for Subtraction.
3. The root cause of treating Parentheses as multiplication symbols is the convention of attaching terms such as xy, that leads to the need of use of grouping symbol to multiply groups of expressions such as Parentheses. Then, Parentheses are treated as multiplication symbols only for grouped expressions. Thus, we must be put a multiplication symbol before or after the parentheses to avoid ambiguity.
4.  The four rules of PEMDAS 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.
5. Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
6. To simplify expressions with negative terms extracted from the parentheses, we need to follow the rule of multiplying integers with different symbols to operate with the preceding term.
7. PEMDAS rule can be applied to real-life situations that involve multiple operations in just one calculation.
8. 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.