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# Numerical Expression & Numerical Equation

Jillian's grandfather asks her a question.

At the start of the month, there were total 115 employees in a company. 40 employees were in management department and rest were in human resource and finance departments. If during the month, the employees in management department got reduced to half, while the employees in human resource and finance departments got doubled, calculate the total employees in the company at the end of the month.

Jillian solved it as follows:

$$40 \div 2 + 75 × 2$$

$$= 20 + 75 × 2$$

$$= 95 × 2$$

$$= 190$$

To determine the correct answer, we need to know about the order of operations.

• For evaluation, we deal with numerical expressions and equations.
• But, what is the difference between them? Let's try to understand these.

Numerical expression

• A numerical expression is a mathematical sentence which has only numbers and one or more operation symbols in it.
• It does not have an equals (=) sign but can be simplified and / or evaluated.
• For example:- $$11 + 9 × 2 + 7$$

Numerical equation

• A numerical equation is a mathematical sentence which tells us that two things are equal.
• To show the equality, it has an equals sign " = ".
• For example:- $$2 + 7 = 9$$

It is an equation which represents that $$2 + 7$$ is equal to $$9$$.

#### Which one of the following is a numerical expression?

A $$25 × 4 = 100$$

B $$25 × 4 = 200 \div 2$$

C $$30 0-200 \div 2 + 10$$

D $$11+12 = 23$$

×

A numerical expression has only numbers and one or more operation symbols.

It does not have an equals sign ( = ), but it can be evaluated.

Options (A), (B), and (D) have equals sign ( = ) which represent that these are equations.

Thus, options (A), (B), and (D) are incorrect.

In option (C), there are only numbers and operations.

Thus, option (C) is correct.

### Which one of the following is a numerical expression?

A

$$25 × 4 = 100$$

.

B

$$25 × 4 = 200 \div 2$$

C

$$30 0-200 \div 2 + 10$$

D

$$11+12 = 23$$

Option C is Correct

# MDAS Rule

• The order of operation is a way of evaluating mathematical statements.
• It is an essential concept to be learnt so that a mathematical statement can be read and solved in the same way by everyone.
• Consider the following expression: $$2 × 8 + 5 - 4 \div 2$$

There are many different ways to evaluate this. Some of them are shown below:

(1)  $$2 × 8 + 5 - 4 \div 2$$

$$= 16 + 5 - 4 \div 2$$

$$= 21 - 4 \div 2$$

$$= 17 \div 2$$

$$= 17 /2$$

(2)  $$2 × 8 + 5 -4 \div 2$$

$$= 2 × 13- 4 \div 2$$

$$=26 - 4 \div 2$$

$$=22 \div 2$$

$$= 11$$

(3)  $$2 × 8 + 5 - 4 \div 2$$

$$= 2 × 8 + 1 \div 2$$

$$= 2 × 9 \div 2$$

$$= 18 \div 2$$

$$=9$$

• It is observed that each method leads to a different solution.
• To evaluate the expression in a specific order so that each person gets the same solution, the MDAS rule is used.

MDAS (Multiplication, Division, Addition, Subtraction) rule

• The MDAS rule states that while evaluating an expression or equation, remember the following points:
• First evaluate multiplication or division and then addition or subtraction.
• MD - Multiplication or division is to be done in the order from left to right.
• AS - Addition or subtraction is to be done in the order from left to right.
• Now, we will apply the MDAS rule in our previous example.

$$2 ×8 + 5- 4 \div 2$$

(1) Multiplication is to be done

$$= 16 + 5 - 4 \div 2$$

(2) Division is to be done

$$= 16 + 5 - 2$$

(3) Addition is to be done

$$= 21 -2$$

(4) Subtraction is to be done

$$= 19$$

• By following the MDAS rule, solution will be the same at each time.

Note:- If any operation is missing, then check for the next order.

#### Choose the option that has the correct order to evaluate the following expression: $$32 \div 4 × 4 - 6 + 12$$

A $$8 × 4 - 6 + 12 \;=\;32-6 +12 = 26 +12 = 38$$

B $$8 × 4 - 6 + 12\;=\; 8×4+6=8×10=80$$

C $$32\div16 - 6 +12\; =\; 2-6 + 12 = -4+12 = 8$$

D $$32 \div 16 - 6 + 12\;=\; 32 \div 10 + 12 = 44 \div 10 = \dfrac{44}{10}$$

×

The given expression consists of all the four operations, i.e. addition, subtraction, multiplication, and division.

Thus, we will use the MDAS rule to evaluate it.

$$32 \div 4 × 4 - 6 + 12$$

(i) Since division is appearing before the multiplication on moving from left to right, so it (division) is to be done first.

$$=8×4-6+12$$

$$8×4-6+12$$

(ii) Multiplication is to be done.

$$=32-6+12$$

$$32-6+12$$

(iii) Since subtraction is appearing before the addition on moving from left to right, so it (subtraction) is to be done first.

$$=26+12$$

$$26+12$$

(iv) Addition is to be done.

$$=38$$

Hence, option (A) is correct.

### Choose the option that has the correct order to evaluate the following expression: $$32 \div 4 × 4 - 6 + 12$$

A

$$8 × 4 - 6 + 12 \;=\;32-6 +12 = 26 +12 = 38$$

.

B

$$8 × 4 - 6 + 12\;=\; 8×4+6=8×10=80$$

C

$$32\div16 - 6 +12\; =\; 2-6 + 12 = -4+12 = 8$$

D

$$32 \div 16 - 6 + 12\;=\; 32 \div 10 + 12 = 44 \div 10 = \dfrac{44}{10}$$

Option A is Correct

# Evaluation of Parentheses

## Parentheses

• Parentheses are also known as brackets.
• There can be three types of brackets, which are:
• (i) Round or curved brackets, denoted by (  )
• (ii) Curly brackets, denoted by {  }
• (iii) Square brackets, denoted by [  ]
• Parentheses are a type of grouping symbols.
• They can be used to show the multiplication of numbers in one of the following ways:

$$1. \;\;\;2(6) = 12\\ 2.\;\;\;(2)(6) = 12\\ 3.\;\;\;[2](6) = 12$$

### Evaluation of parentheses

• When two or more operation symbols are used in parentheses, then they should be evaluated by using the MDAS rule (from left to right).

MDAS rule

M stands for multiplication ( × )

D stands for division ( $$\div$$ )

A stands for addition ( + )

S stands for subtraction ( – )

• Consider the following example:

$$(32-50 \;\div\;2 × 5 + 1)$$

Here, the whole expression is in parenthesis, so we shall use the MDAS rule to evaluate it.

$$(32-50 \;\div\;2 × 5 + 1)$$

(i) Division is to be done

$$= 32-25×5 +1$$

(ii) Multiplication is to be done

$$=32-125+1$$

(iii) Subtraction is to be done

$$=-93+1$$

(iv) Addition is to be done

$$=-92$$

• When one parenthesis is inside another one, then the inner one is to be evaluated first using the MDAS rule.
• Consider the following example:

$$(100- (50 × 1 + 4\; \div 2))$$

Here, two parentheses are used. So, we shall evaluate the inner parenthesis first using the MDAS rule.

$$(100- (50 × 1 + 4\; \div 2))\;\; — (1)$$

Inner parenthesis contains $$(50 × 1 + 4 \;\div 2)$$

(i) Multiplication is to be done $$= 50 + 4 \div 2$$

(ii) Division is to be done $$= 50 + 2$$

(iii) Addition is to be done $$= 52$$

• Now evaluate the outer parenthesis

$$(100\; –\; 52)$$         [From (1)]

Here, only subtraction is to be done $$= 48$$

#### Evaluate the following parentheses $$(112 \; \div (5 × 3 - 8)× 8 + 4)$$

A $$5$$

B $$132$$

C $$104$$

D $$7$$

×

$$(112 \div (5 × 3- 8 ) × 8+ 4)\;\;\;— (1)$$

Here, two parentheses are used. So, we shall evaluate the inner parenthesis first.

Inner parenthesis: $$(5 × 3 - 8)$$

(i) Multiplication is to be done $$= 15 -8$$

(ii) Now only subtraction is to be done $$= 7$$

The outer parenthesis is

$$(112 \div 7 × 8 + 4)$$   [From (1)]

(i) Division is to be done

$$=16×8+4$$

(ii) Multiplication is to be done

$$=128+4$$

(iii) Addition is to be done

$$=132$$

Hence, option (B) is correct.

### Evaluate the following parentheses $$(112 \; \div (5 × 3 - 8)× 8 + 4)$$

A

$$5$$

.

B

$$132$$

C

$$104$$

D

$$7$$

Option B is Correct

# Locating the Grouping Symbols (Parentheses)

## Use of grouping symbols (parentheses)

• Parentheses are used to separate a part of any numerical expression.

For example: $$\dfrac{22}{7} × (42 \div 7-3) - 1$$

Here, $$42 \div 7 -3$$ is separated by parentheses.

• The position of the grouping symbols (parentheses) can change the value of a numerical expression.

For example: $$5 + 8 × 2 \div 4$$

Without parentheses, the solution would be:

$$=5 + 8 × 2 \div 4$$

$$= 5 + 16 \div 4$$

$$= 5 + 4$$

$$= 9$$

Using parentheses at $$(5 + 8)$$, the solution would be:

$$= (5 + 8) × 2 \div 4$$

$$= 13 × 2 \div 4$$

$$=26 \div 4$$

$$= \dfrac{13}{2}$$

Using parentheses at $$(5 + 8 × 2)$$, the solution would be:

$$=5 + 8 × 2 \div 4$$

$$= (5 + 8 × 2 ) \div 4$$

$$= (5 + 16) \div 4$$

$$=21 \div 4$$

$$= \dfrac{21}{4}$$

• It can be observed that by altering the position of parentheses, the value of the expression also gets changed.

#### Which one of the following options represents the correct position of parentheses so that the expression, $$100 \div 5 × 6 - 2$$ gives the result as 80?

A $$(10 0\div 5)× 6 - 2$$

B $$100 \div (5 × 6 )- 2$$

C $$100 \div (5 × 6 - 2)$$

D $$100 \div 5 × (6 - 2)$$

×

Evaluating option (A)

$$(100 \div 5) × 6 -2$$

$$=20 × 6 - 2$$

$$=120 - 2$$

$$= 118$$

Here, the value is not 80.

Thus, option (A) is incorrect.

Evaluating option (B)

$$100 \div (5 × 6 ) - 2$$

$$= 100 \div (30) - 2$$

$$= 3.33- 2$$

$$=1.33$$

Here, the value is not 80.

Thus, option (B) is incorrect.

Evaluating option (C)

$$100 \div (5 × 6 - 2)$$

$$= 100 \div (30- 2)$$

$$= 100 \div 28$$

$$= 3.57$$

Here, the value is not 80.

Thus, option (C) is incorrect.

Evaluating option (D)

$$100 \div 5 × (6-2)$$

$$=100\div 5 × 4$$

$$=20×4$$

$$=80$$

Here, the value is $$80$$.

Thus, option (D) is correct.

### Which one of the following options represents the correct position of parentheses so that the expression, $$100 \div 5 × 6 - 2$$ gives the result as 80?

A

$$(10 0\div 5)× 6 - 2$$

.

B

$$100 \div (5 × 6 )- 2$$

C

$$100 \div (5 × 6 - 2)$$

D

$$100 \div 5 × (6 - 2)$$

Option D is Correct

# PEMDAS Rule

• PEMDAS rule is an order to evaluate any numerical expression containing two or more than two operations.
• PEMDAS rule is as follows:

(i) P stands for parentheses ( )

(ii) E stands for exponents    ^

(iii) M stands for multiplication ×

(iv) D stands for division $$\div$$

(v) A stands for addition +

(vi) S stands for subtraction –

It is clear from above that the parentheses are to be evaluated first, next the exponents, then the multiplication or the division in the order from left to right and at last the addition or the subtraction in the order from left to right.

• In any expression, if one of the operations is missing then we will evaluate the next one.
• PEMDAS rule can be remembered easily by the sentence.

"Please Excuse My Dear Aunt Sally"

• Consider the following example: $$3^3× 1 - (8 \div 2) + 4$$

Here, all the operations are used, thus, we shall follow the PEMDAS rule to evaluate.

$$3^3 × 1 - (8 \div 2) + 4$$

(i) Here, only one pair of parenthesis is used so, we shall evaluate it first.

$$8 \div 2 = 4$$

(ii) Now, the expression is  $$3^3 × 1 - 4 + 4$$

Now, the exponent is to be evaluated.

$$= 27 × 1 - 4 + 4$$

(iii) Multiplication is to be done.

$$= 27 - 4 + 4$$

(iv) Subtraction is to be done.

$$=23+4$$

(v) Addition is to be done.

$$= 27$$

Note: By not following the PEMDAS rule, we would get different and incorrect answers.

#### Evaluate the expression:  $$35 + 3^2 - (3× 2 ) × 7$$

A $$3$$

B $$2$$

C $$4$$

D $$5$$

×

$$35 + 3^2 - (3 × 2) × 7$$

We shall use the PEMDAS rule to evaluate this expression.

$$35 + 3^2 - (3 × 2) × 7$$

(i) Evaluating parenthesis

$$= 35 +3^2 - 6 × 7$$

(ii) Evaluating exponents

$$= 35 + 9 - 6 × 7$$

(iii) Multiplication

$$= 35 + 9- 42$$

$$= 4 4-42$$

(v) Subtraction

$$= 2$$

Hence, option (B) is correct.

### Evaluate the expression:  $$35 + 3^2 - (3× 2 ) × 7$$

A

$$3$$

.

B

$$2$$

C

$$4$$

D

$$5$$

Option B is Correct

# Formation of Numerical Expression

## Numerical expression

• A numerical expression is a mathematical sentence which has only numbers and one or more operation symbols in it.
• It does not have an equals (=) sign but can be simplified and/or evaluated.
• For example:-  $$11 + 9 × 2 + 7$$
• In numerical expressions, mainly four mathematical operations $$(+, \;–, \;×, \;\div)$$, exponents and parentheses are used.
• To evaluate such expressions, we use the PEMDAS rule.
• Now, it's time to learn how to form mathematical expressions from word problems.
• We can use parentheses for our ease.
• Consider the following example:

Kevin has 3 candies and he gets 6 more from his father. Kevin's friend, Sam has 9 candies and he gives 2  to his younger brother. How many candies do both Kevin and Sam have in total?

Here, Kevin has 3 candies and he gets 6 more from his father.

Total candies with Kevin $$= (3 + 6)$$

We will put parenthesis to separate the expression.

Sam has 9 candies and he gives 2  to his younger brother.

Total candies left with Sam $$=(9-2)$$

We need to evaluate the total candies that both have, so we will add.

$$(3 + 6)+ (9- 2)$$

We will evaluate both the parentheses separately.

$$=9 + 7$$

Now the sum $$= 16$$

Thus, both have $$16$$ candies in all.

#### At the start of the month, there were total 115 employees in a company. 40 employees were in management department and rest were in human resource and finance departments. If during the month, the employees in management department got reduced to half, while the employees in human resource and finance departments got doubled, calculate the total employees in the company at the end of the month.

A $$50$$

B $$220$$

C $$170$$

D $$100$$

×

Total employees = $$115$$

Employees in management department = $$40$$

Thus, employees in human resource and finance departments = $$(115-40)$$

During the month,

Employees in management department reduced to half i.e. $$40 \div 2$$

and employees in human resource and finance departments got doubled i.e. $$(115-40)× 2$$

Thus, at the end of the month, total employees were

$$(40 \div 2) + (115- 40 )× 2$$

$$= 20 + 75 × 2$$

$$= 20 + 150$$

$$= 170$$

Hence, option (C) is correct.

### At the start of the month, there were total 115 employees in a company. 40 employees were in management department and rest were in human resource and finance departments. If during the month, the employees in management department got reduced to half, while the employees in human resource and finance departments got doubled, calculate the total employees in the company at the end of the month.

A

$$50$$

.

B

$$220$$

C

$$170$$

D

$$100$$

Option C is Correct