Informative line

Variables (Literals)

• Variables are the alphabets or letters used to represent unknown numbers/unknown quantities.
• These are also known as literals.
• Generally, we use $$x$$ or $$y$$ to represent unknown quantities, but any letter can be used as a variable.

For example: $$a,\;m,\;z,$$ etc can be used as variables.

• Variables are mostly written in lower case.
• Sometimes, figures like $$\Box,\;\Delta,$$ etc are used in place of variables.

Choose the variable in $$5x+10$$

A $$5$$

B $$10$$

C $$x$$

D $$+$$

×

Given: $$5x+10$$

Here, $$x$$ is used to represent an unknown number.

Thus, $$x$$ is a variable.

Hence, option (C) is correct.

Choose the variable in $$5x+10$$

A

$$5$$

.

B

$$10$$

C

$$x$$

D

$$+$$

Option C is Correct

Expression

• A variable can be used in any sort of mathematical expression.

Mathematical Expression

• An expression having numbers with one or more operations and no equals sign is called mathematical expression.

Example: $$2(8+6)$$

Variable Expression/Algebraic Expression

• An expression having variables with one or more operations along with numbers and no equals sign is called variable expression/algebraic expression.

Example: $$6x+7,\;2z,\;3y-3$$ etc.

Which one of the following is an algebraic expression?

A $$3(4+7)$$

B $$71(21\div7)$$

C $$2x+1$$

D $$27(21-17)$$

×

An algebraic expression contains variables, with one or more operations along with numbers and does not contain an equals sign.

Options (A), (B) and (D) have numbers but no variables, so all three are mathematical expressions.

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

Option (C) has variable $$x$$ and numbers are connected with operations, so it is an algebraic expression.

Hence, option (C) is correct.

Which one of the following is an algebraic expression?

A

$$3(4+7)$$

.

B

$$71(21\div7)$$

C

$$2x+1$$

D

$$27(21-17)$$

Option C is Correct

• A variable can be added to numbers.
• The addition is shown by a '$$+$$' sign in between a variable and a number.

For example: $$x+5$$ can be written as

"A variable is added to $$5$$"

or

"Sum of a variable and $$5$$"

or

"$$5$$ more than a variable"

• The addition is shown by using the keywords: 'added', 'sum' or 'more than'.

Which option represents the expression: $$x+10$$?

A $$10$$ more than $$x$$

B $$10$$ less than $$x$$

C $$10$$ times $$x$$

D $$x$$ less than $$10$$

×

The addition is shown by using the keywords: 'added', 'sum' or 'more than'.

In option (A), 'more than' is used which shows addition.

Thus, $$10$$ more than $$x$$ represents $$x+10$$.

Hence, option (A) is correct.

In option (B), 'less than' is used which shows subtraction.

Thus, $$10$$ less than $$x$$ represents $$x-10$$.

Hence, option (B) is incorrect.

In option (C), 'times' is used which shows multiplication.

Thus, $$10$$ times $$x$$ represents $$10x$$.

Hence, option (C) is incorrect.

In option (D), 'less than' is used which shows subtraction.

Thus, $$x$$ less than $$10$$ represents  $$10-x$$.

Hence, option (D) is incorrect.

Which option represents the expression: $$x+10$$?

A

$$10$$ more than $$x$$

.

B

$$10$$ less than $$x$$

C

$$10$$ times $$x$$

D

$$x$$ less than $$10$$

Option A is Correct

Subtraction of Variables and Numbers

• A literal can be subtracted from a number and vice-versa.
• The subtraction is shown by using a '$$-$$' sign between a variable and a number.

Example: $$x-5$$ can be written as

"$$5$$ less than a variable"

or

"$$5$$ subtracted from $$x$$"

• The subtraction is shown by using the keywords: 'taken away', 'less than' or 'subtracted'.

Note: Always take care that $$5$$ subtracted from $$x$$ represents $$(x-5)$$. It is not same as $$x$$ subtracted from $$5$$ which is $$(5-x)$$.

Which option represents the expression: $$x-7$$?

A $$7$$ less than $$x$$

B $$7$$ more than $$x$$

C $$x$$ less than $$7$$

D $$7$$ times $$x$$

×

The subtraction is shown by using the keywords: 'taken away', 'less than' or 'subtracted'.

In option (A), 'less than' is used which shows subtraction.

Thus, $$7$$ less than $$x$$ represents $$x-7$$.

Hence, option (A) is correct.

In option (B), 'more than' is used which shows addition.

Thus, $$7$$ more than $$x$$ represents $$x+7$$.

Hence, option (B) is incorrect.

In option (C), 'less than' is used which shows subtraction.

Thus, $$x$$ less than $$7$$ represents $$7-x$$ which is not same as $$x-7$$.

Hence, option (C) is incorrect.

In option (D), 'times' is used which shows multiplication.

Thus, $$7$$ times $$x$$ represents $$7x$$.

Hence, option (D) is incorrect.

Which option represents the expression: $$x-7$$?

A

$$7$$ less than $$x$$

.

B

$$7$$ more than $$x$$

C

$$x$$ less than $$7$$

D

$$7$$ times $$x$$

Option A is Correct

Multiplication of Variables with Numbers

• A variable can be multiplied with numbers.
• The multiplication is shown either by  '$$.$$' or  '$$×$$' sign between them.

For example: $$7x$$ can be written as

"$$7$$ times $$x$$"

or

"$$7$$ multiplied by $$x$$"

or

"Product of $$7$$ and $$x$$"

• The multiplication is shown by using the keywords: 'times', 'multiplied' or 'product'.

Note: By commutative rule, $$xy$$ is same as $$yx$$.

Which option represents $$20x$$?

A $$20$$ times $$x$$

B $$20$$ divides $$x$$

C $$x$$ divides $$20$$

D $$x$$ more than $$20$$

×

The multiplication is shown by using the keywords: 'times', 'multiplied' or 'product'.

In option (A), 'times' is used which shows multiplication.

Thus, $$20$$ times $$x$$ represents $$20x$$.

Hence, option (A) is correct.

In option (B), 'divides' is used which shows division.

Thus, $$20$$ divides $$x$$ represents $$\dfrac{x}{20}$$.

Hence, option (B) is incorrect.

In option (C), 'divides' is used which shows division.

Thus, $$x$$ divides $$20$$ represents $$\dfrac{20}{x}$$.

Hence, option (C) is incorrect.

In option (D), 'more than' is used which shows addition.

Thus, $$x$$ more than $$20$$ represents $$20+x$$.

Hence, option (D) is incorrect.

Which option represents $$20x$$?

A

$$20$$ times $$x$$

.

B

$$20$$ divides $$x$$

C

$$x$$ divides $$20$$

D

$$x$$ more than $$20$$

Option A is Correct

Division involving Variables and Numbers

• A variable can be divided by a number and vice versa.

For example: $$\dfrac{10}{x}$$ can be represented as

"$$10$$ divided by  $$x$$"

or

"$$10$$ by $$x$$"

Note: It should be taken care that $$10$$ by $$x$$ represents $$\left(\dfrac{10}{x}\right)$$. It is not same as $$x$$ by 10 which is $$\left(\dfrac{x}{10}\right)$$.

• The division is shown by using the keywords: 'by', 'divided by' or 'divides'.

Which option represents the expression: $$\dfrac{x}{10}$$?

A $$x$$ divided by $$10$$

B $$10$$ divided by $$x$$

C $$10$$ times $$x$$

D $$x$$ more than $$10$$

×

The division is shown by using the keywords: 'by', 'divided by' or 'divides'.

In option (A), 'divided by' is used which shows division.

Thus, $$x$$ divided by $$10$$ represents $$\dfrac{x}{10}$$.

Hence, option (A) is correct.

In option (B), 'divided by' is used which shows division.

Thus, $$10$$ divided by $$x$$ represents $$\dfrac{10}{x}$$.

Hence, option (B) is incorrect.

In option (C), 'times' is used which shows multiplication.

Thus, $$10$$ times $$x$$ represents $$10x$$.

Hence, option (C) is incorrect.

In option (D), 'more than' is used which shows addition.

Thus, $$x$$ more than $$10$$ represents $$10+x$$.

Hence, option (D) is incorrect.

Which option represents the expression: $$\dfrac{x}{10}$$?

A

$$x$$ divided by $$10$$

.

B

$$10$$ divided by $$x$$

C

$$10$$ times $$x$$

D

$$x$$ more than $$10$$

Option A is Correct