## Definition of Expression in Math?

An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division. The structure of an expression is:

Expression is (Number/variable, Math Operator, Number/variable)

**Expression Examples:**

Example 1: | $7 + 9$ |

Example 2: | $23.5 \times 4$ |

Example 3: | $37\text{s}$ $–$ $6\text{t}$ |

Example 4: | $25\text{a}^4 + 9$ $–$ $4 ÷ 15$ |

In all the given expressions, a math operator is used between the two numbers.

**Non Examples of Expressions:**

Example 1: *a*

Example 2: $4$

Example 3: $7.89$

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**Parts of an Expression in Math**

An expression in Math is made up of the following:

a) **Constant: **it** **is a fixed numerical value.

Example: $7, 45, 4\frac{1}{3}, −18, \sqrt{5}, 7 + \sqrt{11}$

b) **Variables: **they do not take any fixed values. Values are assigned according to the requirement.

Example: *a, p, z*

c) **Terms:** can be constants, variables or constants multiplied by variable/(s). Each term in an expression is separated by ‘+’ sign or ‘−’ sign

Example: In $5\text{a} + 2\text{b}$ $-$ $7$ the terms are: $5\text{a}, 2\text{b}$, and $7$.

d) **Operators:** The four operations of addition (+) , subtraction (−),multiplication (×), division (÷) are used to combine the terms of an expression and are called operators.

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**Types of Expression in Math**

**Numerical Expression**

Numerical expression in Math consists of numbers and arithmetic operators. It does not contain any unknown variables, equality or inequality symbols.

**Examples:**

$65 + 9$ $–$ $4$

$\frac{25}{4}+\frac{36}{5}$

$42 \div 7 \times 4$ $–$ $9 + 7$

$14.5 + 9$

**Algebraic Expression**

An algebraic expression consists of unknown variables, numbers and arithmetic operators. It does not contain any equality or inequality symbols.

**Examples:**

$5z$

$3\text{x}^2 + 5$

$\frac{2}{5}\text{a}$ $-$ $\frac{7}{11}\text{b} + 4.5\text{c}$

$0.2\text{p}^3$ $\text{q}^2 + \frac{2}{5}$ $\text{p}^2$ $\text{q}$

**Types of Algebraic Expressions **

Algebraic expressions are classified on the basis of the number of terms in the expression. The various types of algebraic expressions are:

**Monomial expressions**contain only one term. E.g. $4\text{x}$**Binomial expressions**contain two unlike terms. E.g. $2\text{xy} +x$**Trinomial expressions**contain only three unlike terms. E.g. $3\text{t}2$ $-$ $4\text{t} + 9$**Polynomial expressions**have two or more terms. This includes binomials and trinomials too and all other expressions with four or more terms. E.g. $2\text{x} + 3\text{y} + 5\text{z}; 4\text{t} + 5$ $−$ $4\text{u} + \text{z}$

**Expression vs Equation**

A math expression is different from a math equation.

The difference between expressions and equations is that an **expression** signifies a combination of numbers, variables, and operation symbols whereas an **equation** will always use an equal (=) operator between two math expressions. Also both sides of the “equal to” sign have the same value.

For example,

Expression | Equation |

$22 + 5$ | $22 + 5 = 29$ $–$ $2$ |

$9 \times 5$ | $9 \times 5 = 45$ |

$50 \div 10$ | $45 \div 9 = 50 \div 10$ |

$15 + 7$ $–$ $6$ | $15 + 7$ $–$ $6 = 16$ |

$25 + 7$ | $25 + 7 = 64 \div 2$ |

$20 \times 5$ | $20 \times 5 = 100$ |

**Where are expressions used?**

Expressions help us in solving word problems. Math expressions are formed using the words of a problem.

Let’s consider the following problem as an example:

Let’s consider a word problem and learn how to write expressions in math

1. Tom has to fill a box with oranges and apples. The number of apples should be 5 more than oranges. Tom picks 3 oranges each time and repeats it 5 times. Count the total number of oranges and apples.

To solve this, formulate the math expressions as follows:

Number of oranges = $3 \times 5$

Number of oranges = $15$

Number of apples = Number of oranges $ + 5$

Number of apples = $15 + 5$

Number of apples = $20$

Total number of fruits = Number of oranges + Number of apples

Third math expression will be:

Total number of fruits = $15 + 20$ (Substituting the value of number of oranges and apples)

$= 35$

2. A class of students are going on a trip. Each student has to pay an individual fee of $\$$8 and a group fee of $\$$30. Write an algebraic expression for the total cost of the trip. Find the total cost of the trip if there are 56 students going on the trip.

Let *n* represent the number of students.

Algebraic expression will be:

$\$$8*n* + $\$$30

To find the total cost of the trip, when *n* = 56.

$\$$8*n* + $\$$30

$\$$8(56) + $\$$30 (Substituting *n* with 56)

$\$$448 + $\$$30

$\$$478

**PEDMAS**

PEDMAS is an acronym where P stands for parenthesis, E for exponents, D for division, M for multiplication, A for addition, and S stands for subtraction.

The order of the letters indicate the order in which the operation must be performed.

The operations of multiplication and division have the same level of priority. Always perform the operation which appears first from left to right.

Similarly addition and subtraction have the same level of priority. Here also perform the one that appears first from left to right.

Example:

$(15 \div 3 \times 4$ $−$ $7) + (19$ $−$ $4^2)$

$= (5 × 4 − 7) + × (19 − 16)$

$= (20 − 7) + 3$

$= 13 + 3$

$= 16$

Application:

The knowledge of applying math operations on numbers is the first step towards building basic arithmetic reasoning and logic in children. Formulation of math expressions using the respective skill lays a strong foundation to learn algebra and translate real-life problems in suitable mathematical models.

## Solved Examples on Expression:

**Example 1**: Write whether each is an expression or an equation.

(a) $4 + 8$ | (b) $4 + 12 = 16$ | (c) $5 \times 35$ |

(d) $16 \div 4 + 9$ $–$ $2$ | (e) $8 \times 4 \div 2 = 16$ | (f) $72+94$ |

Solution:

Expression | Equation |

$4 + 8$ | $4 + 12 = 16$ |

$5 \times 35$ | $8 \times 4 \div 2 = 16$ |

$16 \div 4 + 9$ $–$ $2$ | |

$72+94$ |

**Example 2**: Write each word phrase as an expression.

- The sum of $10$ and $14$
- 3 more than a number $7$
- Two times $11$, increased by $1$
- 19 less than the product of $15$ and $4$
- The quotient of $33$ and $3$

**Solution:**

- $10 + 14$
- $7 + 3$
- $2 \times 11 + 1$
- $15 \times 4$ $–$ $19$
- $33 \div 3$

**Example 3**: Classify the following expression as arithmetic or algebraic.

- $4\text{a}$ $–$ $7\text{b}$
- $23 + 42$ $–$ $6$
- $715$ $-$ $911$
- $2$ $-$ $5\text{x}9\text{y}$
- $22$ $–$ $5 + 8$
- $7\text{y} + 19\text{x}$ $–$ $4\text{z}$

**Solution:**

Arithmetic | Algebraic |

$23 + 42$ $–$ $6$ | $4\text{a}$ $–$ $7\text{b}$ |

$22$ $–$ $5 + 8$ | $7\text{y} + 19\text{x}$ $–$ $4\text{z}$ |

$715$ $-$ $911$ | $2$ $-$ $5x9y$ |

**Example 4: Write the terms of the given expression **$4uv + 7u − 9z + 6z$**.**

**Solution:**

$4\text{uv}, 7\text{u}$, $−$ $9\text{z}$ and $6\text{z}$ are terms of the given expression.

**Example 5: **A book has $250$ pages. Ron has $62$ pages left to read. Write an expression to find the number of pages he has read.

**Solution:**

$250$ $–$ $62$

**Example 6:** $X$, $Y$, and $Z$ have a few hairbands. $Y$ has $20$ more hairbands than $X$. $Z$ says that she has five more hairbands than the number of headbands that $X$ and $Y$ together have. Express this in the form of an expression?

**Solution**: Let the number of hair bands with $X$ be$ = \text{x}$.

Then, $Y$ has $(\text{x}+20)$ hairbands.

$Z$ has $\text{x} + (\text{x}+20) + 5=2\text{x}+25$ hairbands.

Therefore, $Z$ has $(2\text{x}+25)$ hairbands.

## Practice Problem

## Expression in Maths – Definition with Examples￼

### Which of these is an expression?

As it has numbers (fraction) and mathematical operators, other options are equations as it has equal $(=)$ operator between two math expressions.

### Which of these is algebraic expressions?

It contains variables, numbers, and mathematical operator.Option (a) is an equation, option (c) and option (d) are arithmetic expressions not algebraic as no variables are involved.

### Any algebraic expression containing only three terms would be called?

If there are three terms in an expression, we call it trinomial expression.

### If the value of a term in an expression does not change, what is it called?

The term that does not change its value is called a constant.

### Write the the expression for this word phrase: ‘Three times a number 14, decreased by 10

Three times a number 14 is $3 \times 14$.

This decreased by 10 equals $3 \times 14$ $−$ $10$

### Thomas earns $\$$9.75 an hour as a librarian. Which of these expressions shows how much he earns in 40 hours?

Amount earned in an hour $=$ $\$$9.75

Amount earned in 40 hours $=$ $\$$$(9.75 \times 40)$

## Frequently Asked Questions

What is an expression in maths?

An expression is a set of numbers or variables combined using the operations $+$, $–$, $\times$ or $\div$.

What are types of expression?

Arithmetic expression that contains only numbers and mathematical operators and algebraic expression that contains variables, numbers and mathematical operators.

Can we solve a math expression?

No, we cannot solve a math expression as it does not have an ‘equal to’ sign $( = )$ but we can simplify expressions.

What is the difference between arithmetic expression and algebraic expression?

Mathematical expressions have only numbers and operators, while algebraic expressions have both numbers and variables in terms, separated by operators in between.