Algebraic Expressions

Math, like any other language, has a way of communicating ideas. An algebraic expression is a concise way of describing mathematical objects through the use of numbers, variables (letters), and arithmetic operations such as addition, subtraction, multiplication, and division.

The three main components of algebraic expressions are numbers, variables, and arithmetic operations.

  • Numbers or Constants

Examples: [latex]1[/latex], [latex]6[/latex], [latex]8[/latex], [latex]27[/latex], [latex]32[/latex], etc.

  • Variables or Letters

Examples: [latex]x[/latex], [latex]y[/latex], [latex]a[/latex], [latex]h[/latex], [latex]p[/latex], etc.

  • Arithmetic Operations

Examples: [latex]+[/latex] (addition),  [latex] – [/latex] (subtraction) ,  [latex] \times[/latex] (multiplication) , [latex]÷[/latex] (division)

The following are easy examples that can help you get familiar with the operations of addition, subtraction, multiplication, and division.

  • Addition

the sum of [latex]x[/latex] and [latex]5[/latex] → [latex]x+5[/latex]

  • Subtraction

the difference of [latex]y[/latex] and [latex]3[/latex] → [latex]y-3[/latex]

  • Multiplication

the product of [latex]n[/latex] and [latex]2[/latex] → [latex]2n[/latex]

  • Division

the quotient of [latex]k[/latex] and [latex]7[/latex] → [latex]\Large{{k \over 7}}[/latex]

Writing Algebraic Expressions Step-by-Step Examples

Let’s go over more examples.

Example 1: The sum of twice a number and [latex]3[/latex]

Answer: Let variable [latex]x[/latex] be the unknown number. So twice a number means [latex]2x[/latex]. The sum (use plus symbol) of twice a number and [latex]3[/latex] can be written as [latex]2x+3[/latex].

Example 2: The difference of triple a number and [latex]5[/latex]

Answer: Let variable [latex]y[/latex] be the unknown number. So triple a number means [latex]3y[/latex]. The difference (use minus symbol) of triple a number and [latex]5[/latex] should be written as [latex]3y – 5[/latex].

Example 3: The sum of the quotient of [latex]m[/latex] and [latex]2[/latex], and the product of [latex]4[/latex] and [latex]n[/latex].

Answer: In this case, the unknown numbers are already provided as [latex]m[/latex] and [latex]n[/latex]. That’s one less thing to worry about.

The key is to recognize that we are going to add a quotient and a product.

  • the quotient of [latex]m[/latex] and [latex]2[/latex] is expressed as [latex]\Large{{m \over 2}}[/latex]
  • the product of [latex]4[/latex] and [latex]n[/latex] is expressed as [latex]4n[/latex]

Therefore, the sum of the quotient and product is [latex]{\Large{{m \over 2}}} + 4n[/latex].

Example 4: The difference of the product of [latex]7[/latex] and [latex]w[/latex], and the quotient of [latex]2[/latex] and [latex]v[/latex].

Answer: In this case, the unknown numbers have been assigned with corresponding variables which are [latex]w[/latex] and [latex]v[/latex].

 The key is to recognize that we are going to subtract the product by the quotient of some expressions.

  • the product of [latex]7[/latex] and [latex]w[/latex] is expressed as [latex]7w[/latex]
  • the quotient of [latex]2[/latex] and [latex]v[/latex] is expressed as [latex]\Large{{2 \over v}}[/latex]

Therefore, the difference of the product and quotient is [latex]7w – {\Large{{2 \over v}}}[/latex].

Common Words or Terms to Mean Addition, Subtraction, Multiplication, and Division

Let’s go over some common words or phrases that describe the four arithmetic operations. It is critical that you understand these words or phrases in order to successfully write or interpret any given algebraic expression.

algebraic terms which imply addition are plus, more than, sum, and, total, greater than, and increased by.
algebraic terms which imply subtraction are minus, difference, take away, subtract, less than, decreased by, and less.
algebraic terms which imply multiplication are multiply, product, double, twice, triple, quadruple, and times.
algebraic terms which imply division are quotient, divided by, ratio, share, into, and over.

Math Phrases into Algebraic Expressions

The key to learning is to study a LOT of examples!

a number plus 9y + 9
the sum of a number and 10m + 10
total of a number and 5b + 5
a number increased by 4x + 4
h take away 2h − 2
2 take away by a number2 − h
a number minus 11k − 11
11 minus a number11 − k
a number decreased by 7y − 7
the difference of n and 25n − 25
the difference of 25 and n25 − n
5 less than a numberx − 5
x less than the number 55 − x
the product of r and 44r
7 times a number7p
double a number2x
triple a number3x
a number divided by 4w / 4
the quotient of w and 6w / 6
the quotient of 12 and m12 / m
a number divided by 3f / 3
t over 7t / 7
5 into a numbera / 5
a number into 55 / a
the sum of x and 7 divided by 2( x + 7 ) / 2
the difference of m and 3 over 5( m − 3) / 5
11 more than the product of 3 and y3y + 11
6 less than the quotient of c and 10( c / 10 ) − 6
3 minus the product of 5 and a number3 − 5x
the sum of 5 and the quotient of z and 7( z / 7 ) + 5
the difference of twice a number and 32m − 3

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Algebraic Expressions Worked Examples

Algebraic Sentences Word Problems