# Algebraic Expressions

Just like any language, math has a way to communicate ideas. An algebraic expression is a compact way of describing mathematical objects using a **combination** of numbers, variables (letters), and arithmetic operations namely addition, subtraction, multiplication, and division.

In other words, the three main components of algebraic expressions are **numbers**, **variables**, and **arithmetic operations**.

**NUMBERS OR CONSTANTS**

Examples: 1, 6, 8, 27, 32, etc.

**VARIABLES OR LETTERS**

Examples: x, y, a, h, p, etc.

**ARITHMETIC OPERATIONS**

Examples: + (addition), - (subtraction) , \times (multiplication) , ÷ (division)

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

**Addition**

the sum of x and 5 → x+5

**Subtraction**

the difference of y and 3 → y-3

**Multiplication**

the product of n and 2 → 2n

**Division**

the quotient of k and 7 → {k \over 7}

## Writing Algebraic Expressions Step-by-Step Examples

Let’s go over more examples.

**Example 1:** The sum of twice a number and 3

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

**Example 2:** The difference of triple a number and 5

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

**Example 3:** The sum of the quotient of m and 2, and the product of 4 and n.

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

The key is to recognize that we are going to **add** a __quotient__ and a __product__.

- the quotient of m and 2 is expressed as {m \over 2}

- the product of 4 and n is expressed as 4n

Therefore, the sum of the quotient and product is {m \over 2} + 4n.

**Example 4:** The difference of the product of 7 and w, and the quotient of 2 and v.

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

The key is to recognize that we are going to **subtract** the __product__ by the __quotient of some expressions__.

- the product of 7 and w is expressed as 7w

- the quotient of 2 and v is expressed as {2 \over v}

Therefore, the difference of the product and quotient is 7w - {2 \over v}.

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

Now, let’s go over some common words or phrases that describe the four arithmetic operations. It is critical that you know these words or phrases to be successful in writing or interpreting any given algebraic expression.

**ADDITION**

plus

more than

sum

and

total

greater than

increased by

**SUBTRACTION**

minus

difference

take away

subtract

less than

decreased by

less

**MULTIPLICATION**

multiply

product

double (multiply by 2)

twice (multiply by 2)

triple (multiply by 3)

quadruple (multiply by 4)

times

**DIVISION**

quotient

divided by

ratio

share

into

over

#### Math Phrases into Algebraic Expressions

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

MATH PHRASES | ALGEBRAIC EXPRESSIONS |

a number plus 9 | y + 9 |

the sum of a number and 10 | m+10 |

total of a number and 5 | b + 5 |

a number increased by 4 | x + 4 |

h take away 2 | h - 2 |

2 take away by a number | 2 - h |

a number minus 11 | k - 11 |

11 minus a number | 11 - k |

a number decreased by 7 | d - 7 |

the difference of n and 25 | n - 25 |

the difference of 25 and n | 25 - n |

5 less than a number | x - 5 |

x less than the number 5 | 5 - x |

the product of r and 4 | 4r |

7 times a number | 7p |

double a number | 2x |

triple a number | 3x |

a number divided by 4 | {w \over 4} |

the quotient of x and 6 | {x \over 6} |

the quotient of 12 and m | {{12} \over m} |

a number divided by 3 | {f \over 3} |

y over 7 | {y \over 7} |

5 into a number | {a \over 5} |

a number into 5 | {5 \over a} |

the sum of x and 7 divided by 2 | {{x + 7} \over 2} |

the difference of m and 3 over 5 | {{m - 3} \over 5} |

11 more than the product of 3 and y | 3y + 11 |

6 less than the quotient of c and 10 | {c \over {10}} - 6 |

3 minus the product of 5 and a number | 3 - 5x |

the sum of 5 and the quotient of z and 7 | {z \over 7} + 5 |

the difference of twice a number and 3 | 2m - 3 |