Informative line

Transformation Of Expressions

Transformation of a Phrase into an Expression (two or more operations together)

  • A verbal model or phrase having keywords for different operations can be converted into an algebraic expression using suitable operations.
  • We can use the following different options for keywords.

For addition: Plus, more than, added to, and, sum, altogether etc.

For subtraction: Less, less than, fewer than, subtracted, minus, take away, difference, spent etc.

For multiplication: Product, times, multiplied, twice \((2×)\) etc.

For division: Shared equally, divided, quotient of, spilt up, distributed, ratio, half etc.

  • Consider an example to understand it easily.

\(7\) divided by the sum of twice of a number and \(6.\)

  • Consider the verbal model: Twice of a number.

Here, keyword 'twice of' is representing 'multiplication by \(2\)' and 'a number' is representing a 'variable', say, \(x.\)

Thus, the expression for 'twice a number' is:

\(2×x\)

or

\(2x\)

  • Consider the verbal model: the sum of twice of a number and \(6.\)

Here, keyword 'sum' is representing 'addition' operation between 'twice of a number \((2x)\)' and \('6'\).

Thus, expression for 'the sum of twice a number and \(6\)' is

\(2x+6\)

  • Consider the complete verbal model or phrase: \(7\) divided by the sum of twice of a number and \(6.\)

Here, keyword 'divided by' is representing 'division' operation.

Thus, expression for \(7\) divided by the sum of twice of a number and \(6\) is:

\(\dfrac{7}{2x+6}\)

Illustration Questions

Choose the correct expression for the given verbal model: \(3\) less than the product of \(4\) and a number.

A \(4x+3\)

B \(\dfrac{4x}{3}\)

C \(4x-3\)

D \(3x-4\)

×

Consider the verbal model: The product of \(4\) and a number.

Here, keyword 'product' is representing 'multiplication' operation and 'a number' is representing 'variable' say, \(x.\)

Thus, expression for 'the product of \(4\) and a number' is:

\(4×x\;\text{or}\;4x\)

Consider the complete verbal model: \(3\) less than the product of \(4\) and 'a number'.

Here, keyword 'less than' is representing 'subtraction' operation.

Thus, expression for '\(3\) less than the product of \(4\) and a number' is:

\(4x-3\)

Hence, option (C) is correct.

Choose the correct expression for the given verbal model: \(3\) less than the product of \(4\) and a number.

A

\(4x+3\)

.

B

\(\dfrac{4x}{3}\)

C

\(4x-3\)

D

\(3x-4\)

Option C is Correct

Forming Expressions from Real Situations (single variable)

There are many situations in real world that can be written as expressions.

  • Follow the given steps to write an algebraic expression from real situation.

Step 1: Identify the unknown quantity and consider it as a variable.

Step 2: Now read the situation carefully and analyze the keywords for operations to write the expression.

  • We can use the following different options for keywords.

For addition: Plus, more than, added to, and, sum, altogether etc.

For subtraction: Less, less than, fewer than, subtracted, minus, take away, difference, spent etc.

For multiplication: Product, times, multiplied, twice \((2×)\) etc.

For division: Shared equally, divided, quotient of, spilt up, distributed, ratio, half etc.

  • Consider an example to understand it easily.

For example:

Cody had \($\,500\). He spent some money on shopping. How much money does he have now?

\(\to\) Before shopping Cody had money \(=$\,500\)

\(\to\) He spent some money represents an unknown quantity so, consider it as a variable.

Let, he spent \($\,x\) on shopping.

  • We know that 'spent' is the keyword that is used for 'subtraction' operation.

\(\to\) After shopping, there is a shortage of \($\,x\) from \($\,500\). It means \(x\) less than \(500\).

Thus, expression for the given situation is:

\(500-x\)

Illustration Questions

Ms. Wendy purchased two times the same number of chocolates. Next day, she distributed all the chocolates among \(30\) students. Write an expression to represent the given situation.

A \(2x-30\)

B \(\dfrac{2x}{30}\)

C \(\dfrac{2x+2x}{30}\)

D \(\dfrac{30}{2x}\)

×

Here 'number of chocolates' represents an unknown quantity so, consider it as a variable.

Let, number of chocolates \(=x\)

Ms. Wendy purchased two times the same number of chocolates.

We know that 'times' is the keyword that is used for 'multiplication' operation.

\(\therefore\) she purchased \(2x\) chocolates.

Now, she distributed all the chocolates among \(30\) students.

Here 'distributed' is the keyword that is used for 'division' operation.

\(\therefore\) Expression for the whole situation is:

\(\dfrac{2x}{30}\)

Hence, option (B) is correct.

Ms. Wendy purchased two times the same number of chocolates. Next day, she distributed all the chocolates among \(30\) students. Write an expression to represent the given situation.

A

\(2x-30\)

.

B

\(\dfrac{2x}{30}\)

C

\(\dfrac{2x+2x}{30}\)

D

\(\dfrac{30}{2x}\)

Option B is Correct

Conversion of Expressions to Phrases for Single Variable, Single Operation

  •  A variable can be added, subtracted, multiplied or divided by a number.

For example:

(i) \(x+5\)

  • Here, \(x\) is a variable.
  • \(5\) is a known number.
  • \('+'\) represents addition.

Thus, we can write  '\(x\) is added to \(5\)'.

The addition is shown by using keywords:

'added', 'sum', 'more than' etc.

(ii) \(x-5\)

  • Here, \(x\) is a variable.
  • \(5\) is a known number.
  • \('-'\) represents subtraction.

Thus, we can write '\(5\) less than \(x\)'.

The subtraction is shown by using keywords:

'taken away', 'less than', 'subtracted' etc.

(iii) \(7x\)

  • Here, \(x\) is a variable.
  • \(7\) is a known number.
  • Multiplication operation is used.

Thus, we can write '\(7\) times \(x\)'.

The multiplication is shown by using keywords:

'times', 'multiplied', 'product' etc.

(iv) \(\dfrac{10}{x}\)

  • Here, \(x\) is a variable.
  • \(10\) is a known number.
  • Fraction bar represents division.

Thus, we can write '\(10\) divided by \(x\)'.

The division is shown by using keyword:

'by', 'divided by', 'divides', 'quotient' etc.

Illustration Questions

Choose the correct statement for the given expression: \(\dfrac{5}{x}\)

A \(5\) is added to \(x\)

B \(5\) is divided by \(x\)

C \(x\) is divided by \(5\)

D \(5\) times \(x\)

×

Given algebraic expression is \(\dfrac{5}{x}\).

In \(\dfrac{5}{x}\), there is a fraction bar which represents the division operation between \(5\) and \(x.\)

The keyword used for division operation is 'divided by'.

So, we can write:

 \(5\) is divided by \(x.\)

Hence, option (B) is correct.

Choose the correct statement for the given expression: \(\dfrac{5}{x}\)

A

\(5\) is added to \(x\)

.

B

\(5\) is divided by \(x\)

C

\(x\) is divided by \(5\)

D

\(5\) times \(x\)

Option B is Correct

Conversion of Phrases to Expressions for Single Variable, Single Operation

  • Variables can be operated with different operations.
  • Follow the following steps to write an expression for a given statement:

(i) Identify the given number.

(ii) Identify the operation involved.

(iii) Identify the variable.

(iv) Write an expression using variable, numbers and operation.

Example:

(i) 'A variable added to \(5\)'

Here, 'added to' indicates addition operation.

Thus, it can be written as \(x+5\).

The addition is shown by using keywords:

'added', 'sum', 'increase', 'more than' etc.

(ii)  '\(2\) subtracted from \(y\)'

Here, 'subtracted' indicates subtraction.

Thus, it can be written as \(y-2\).

The subtraction is shown by using keywords:

'taken away', 'less than', 'decreased', 'subtracted to' etc.

(iii) '\(4\) times \(x\)'

Here, 'times' indicates multiplication.

Thus, it can be written as \(4x\).

The multiplication is shown by using keywords:

'times', 'multiplied', 'product' etc.

(iv) '\(x\) divided by \(2\)'

Here, 'divided' indicates division.

Thus, it can be written as \(\dfrac{x}{2}\).

 The division is shown by using keywords:

'by', 'divided by', 'divides' etc.

Illustration Questions

Choose the correct expression for the given statement: 'A number is decreased by \(7\)'.

A \(x-7\)

B \(7-x\)

C \(x+7\)

D \(7x\)

×

Given statement is:

'A number is decreased by \(7\)'.

A number which is unknown let it be \(x\) and \(7\) is a known number. Here 'decreased' word shows the subtraction operation.

Thus, Variable \(=x\)

Number \(=7\)

Involved operation = Subtraction

So, we can write 'a number \((x)\) is decreased by \(7\)' as :

 \(x-7\)

Hence, option (A) is correct.

Choose the correct expression for the given statement: 'A number is decreased by \(7\)'.

A

\(x-7\)

.

B

\(7-x\)

C

\(x+7\)

D

\(7x\)

Option A is Correct

Transformation of an Expression into a Phrase

  • An algebraic expression having more than one operations can be transformed into a verbal model or phrase using suitable keywords.
  • We can use the following keywords for different operations.

For addition: Plus, more than, added to, and, sum, altogether etc.

For subtraction: Less, less than, fewer than, subtracted, minus, take away, difference, spent etc.

For multiplication: Product, times, multiplied, twice \((2×)\) etc.

For division:  Shared equally, divided, quotient of, split up, distributed, ratio, half etc.

  • Consider an example to understand it easily.

\(3x+5\)

  • We can transform the given algebraic expression into a verbal model or phrase as following:

\(\to\) Two operations are used in the given expression.

\(\to\) For term \(3x\;\text{or}\;3×x\), we can say as:

\(3\) times \(x.\)

Here 'times' is the keyword for multiplication.

\(\to\) For expression \(3x+5\),

\(5\) is being added to \(3x\) so we can say that it is the 'sum' of \(3x\) and \(5\).

Here 'sum' is the keyword for addition.

\(\to\) Thus we can write a verbal model or phrase for \(3x+5\) as: The sum of three times \(x\) and \(5\).

Illustration Questions

Choose the correct verbal model for the given expression:   \(\dfrac{12-2y}{4}\)

A \(2\) times \(y\) less than \(12\) is divided by \(4\).

B The product of \(2\) and \(y\) divided by \(4\) is less than \(12\).

C The sum of \(2\) and \(y\) less than \(12\) is divided by \(4\).

D The sum of \(2\) and \(y\) divided by \(4\) is less than \(12\).

×

Three operations are used in the given algebraic expression.

For term \(2y,\;y\) is being multiplied by \(2\) so, we can say:

\(2\) times \(y\).

Here, 'times' is the keyword for multiplication.

For terms \(12-2y\),

\(2y\) is being subtracted from \(12\) so, we can say:

\(2y\) less than \(12\).

Here 'less than' is the keyword for subtraction.

Thus, we can write a verbal model or phrase for \(12-2y\) as:

\(2\) times \(y\) less than \(12\).

For expression \(\dfrac{12-2y}{4}\),

\(12-2y\) is being divided by \(4\) so, we can say:

\(12-2y\) is divided by \(4.\)

Here 'divided' is the keyword for division.

Thus, we can write a verbal model or phrase for \(\dfrac{12-2y}{4}\) as:

\(2\) times \(y\) less than \(12\) is divided by \(4\).

Hence, option (A) is correct.

Choose the correct verbal model for the given expression:   \(\dfrac{12-2y}{4}\)

A

\(2\) times \(y\) less than \(12\) is divided by \(4\).

.

B

The product of \(2\) and \(y\) divided by \(4\) is less than \(12\).

C

The sum of \(2\) and \(y\) less than \(12\) is divided by \(4\).

D

The sum of \(2\) and \(y\) divided by \(4\) is less than \(12\).

Option A is Correct

Forming Expressions from Real Situations (Multiple Variables)

There are many situations in the real world that can be written as expressions.

  • Follow the given steps to write an algebraic expression from a real situation.

Step 1: Identify the unknown quantities and consider them as variables.

Step 2: Now read the situation carefully and analyze the keywords for operations to write the expression.

  • We can use the following different options for keywords.

For addition: Plus, more than, added to, and, sum, altogether etc.

For subtraction: Less, less than, fewer than, subtracted, minus, take away, difference, spent etc.

For multiplication: Product, times, multiplied, twice \((2×)\) etc.

For division: Shared equally, divided, quotient of, spilt up, distributed, ratio, half etc.

  • Consider an example to understand it easily.

Example: Ana's monthly earnings, \($\,x\) and \($\,y\) come from baby sitting and home tuition respectively. She saves half of her total earnings for the education of her younger brother.

Write an expression for the amount she saves.

Earnings from baby sitting \(=$\,x\)

Earnings from home tuition \(=$\,y\)

\(\to\) Here, keyword 'total'  is representing the 'addition' of \($\,x\) and \($\,y\).

\(\therefore\) Her total earnings \(=x+y\)

\(\to\) She saves half of her total earnings.

Here, keyword 'half of' is representing 'division by \(2\)' or 'multiplication by \(\dfrac{1}{2}\)'.

\(\therefore\) Expression for the given situation is:

\(\dfrac{x+y}{2}\)

Illustration Questions

Sarah has \(x\) flowers, out of which she gives \(y\) flowers to her mother and twice as many flowers as mother to her father. Write an expression to represent the number of flowers, Sarah is left with.

A \(x-y+2y\)

B \(2y-x-y\)

C \(\dfrac{x}{y+2y}\)

D \(x-(y+2y)\)

×

Given: Sarah had flowers \(=x\)

Flowers given to mother \(=y\)

Here, keyword 'twice' is representing 'multiplication (by \(2\)) operation'.

Expression for, 'twice to her father as compared to her mother is' : \(2y\)

\(\therefore\) Flowers are given to father \(=2y\)

Since, flowers that are given to her mother and father are given from total flowers \(x.\)

\(\therefore\) These are subtracted from \(x.\)

Thus, expression for the number of flowers, Sarah is left with is:

\(x-(y+2y)\)

Hence, option (D) is correct.

Sarah has \(x\) flowers, out of which she gives \(y\) flowers to her mother and twice as many flowers as mother to her father. Write an expression to represent the number of flowers, Sarah is left with.

A

\(x-y+2y\)

.

B

\(2y-x-y\)

C

\(\dfrac{x}{y+2y}\)

D

\(x-(y+2y)\)

Option D is Correct

Real Situations from Expressions

Illustration Questions

Choose the correct option to describe the given expression: \(\dfrac{x+40}{y}\)

A Jacob has \(x\) pencils, purchases \(40\) more pencils and distributes all the pencils among \(y\) students.

B Jacob has \(y\) pencils, purchases \(x\) more pencils and gives away \(40\) pencils to a little girl.

C Jacob has \(x\) pencils, purchases \(y\) more pencils and distributes all the pencils among \(40\)  students.

D Jacob has \(x\) pencils, \(40\) pencils are useless and distributes rest of the pencils among \(y\) students.

×

Option (A):

Given: Jacob has \(x\) pencils.

He purchases \(40\) more pencils.

Here, 'more' is the keyword for 'addition' operation.

\(\therefore\) Total pencils Jacob has \(=x+40\)

Now, he distributes all the pencils among \(y\) students.

Here, 'distributes' is the keyword for 'division' operation.

\(\therefore\) Expression is: \(\dfrac{x+40}{y}\)

Hence, option (A) is correct.

Option (B):

Given: Jacob has \(y\) pencils.

He purchases \(x\) more pencils.

Here, 'more' is the keyword for 'addition' operation.

\(\therefore\) Total pencils Jacob has \(=y+x\)

Now, he gives away \(40\) pencils to a little girl.

Here, 'gives away' is the keyword for 'subtraction' operation.

\(\therefore\) Expression is: \(y+x-40\)

Hence, option (B) is incorrect.

Option (C):

Given: Jacob has \(x\) pencils.

He purchases \(y\) more pencils.

Here, 'more' is the keyword for 'addition' operation.

\(\therefore\) Total pencils Jacob has \(=x+y\)

Now, he distributes all the pencils among \(40\) students.

Here, 'distributes' is the keyword for 'division' operation.

\(\therefore\) Expression is: \(\dfrac{x+y}{40}\)

Hence, option (C) is incorrect.

Option (D):

Given: Jacob has \(x\) pencils.

Useless pencils \(=40\)

Now, he distributes rest of the pencils among \(y\) students.

Here, 'rest' is indicating that \(40\) pencils are 'fewer than' \(x\) and 'fewer than' is the keyword for 'subtraction' operation.

\(\therefore\) Rest of the pencils \(=x-40\)

Here, 'distributes' is the keyword for 'division' operation.

\(\therefore\) Expression is: \(\dfrac{x-40}{y}\)

Hence, option (D) is incorrect.

Choose the correct option to describe the given expression: \(\dfrac{x+40}{y}\)

A

Jacob has \(x\) pencils, purchases \(40\) more pencils and distributes all the pencils among \(y\) students.

.

B

Jacob has \(y\) pencils, purchases \(x\) more pencils and gives away \(40\) pencils to a little girl.

C

Jacob has \(x\) pencils, purchases \(y\) more pencils and distributes all the pencils among \(40\)  students.

D

Jacob has \(x\) pencils, \(40\) pencils are useless and distributes rest of the pencils among \(y\) students.

Option A is Correct

Practice Now