# Translating Basic Math Phrases into Algebraic Expressions

There is no single strategy how to translate math phrases into algebraic expressions. As long as you can remember the basics, you should be able to tackle the more challenging ones. Just make sure that you can justify how you come up with your own algebraic expression, and more importantly that it makes sense to you. Always ask help from your teachers, as needed or collaborate with your classmates so that you can verify your answers.

To build your skills on writing algebraic expressions, we will go over different ways how each operation may show up as a word or phrase in the problem. The four arithmetic operations involved are addition, subtraction, multiplication, and division.

## Key Words for Addition

## Key Words for Subtraction

## Key Words for Multiplication

## Key Words for Division

It is time now time to go over some examples to practice writing algebraic expressions. Consider the following problems as “basic examples” because the algebraic expressions will only involve a single operation. For more advanced or slightly complicated expressions, we have a separate lesson for that which is writing multi-part algebraic expressions.

### Examples of How to Translate Basic Math Phrases into Algebraic Expressions

We will go over eight (8) examples in this lesson to accommodate two (2) two examples for each operation.

**Example 1**: Write an algebraic expression for the math phrase ” the sum of a number and four”.

** Solution**: The word “sum” immediately gives us the hint that we are going to add here. Notice that we want to add two quantities: one unknown number and the number

**4**. Since we don’t know what the value of the number, we can use a variable to represent it. You may use any letters of the alphabet. In this case, let’s agree to use

*for the variable.*

**y**When we add the variable * y* and

**4**, we have

**y +****4**. It is also okay to write your answer as

**4 +**because addition is commutative – that is, switching the order of addition doesn’t change its sum.

*y***Example 2**: Write an algebraic expression for the math phrase ” 10 increased by a number”.

** Solution**: The key words “increased by” implies addition. This means that an unknown number has been added to

**10**. Using the letter

*as the variable, we can translate the statement above as*

**k****10 +**. Since addition is commutative, we can rewrite it as

*k***. Either of the two above is a correct answer.**

*k*+ 10**Example 3**: Write an algebraic expression for the math phrase ” the difference of 1 and a number”.

** Solution**: The word “difference” suggests that we are going to subtract. In addition, when you encounter this math word (difference) make sure to pay attention to the order. The number 1 comes first then an unknown number comes in second. That means the

**number 1**is the

**minuend**and the

**unknown number**is the

**subtrahend**. If we decide to the use the letter

*as our variable, the answer becomes*

**x****1 −**.

*x***Example 4**: Write an algebraic expression for the math phrase ” a number less than 8″.

** Solution**: Be very careful when dealing with the keywords “less than”. The first quantity that comes before the “less than” keywords which is ” a number” is the subtrahend. While the quantity that comes after it becomes the minuend.

In other words, we are going to subtract the unknown number from the number 8. If we choose our variable to be the letter* a*, we get

**8 −**.

*a***Example 5**: Write an algebraic expression for the math phrase ” the product of 5 and a number”.

** Solution**: To find the product of two quantities or values, it means that we will multiply them together. Selecting the letter

*as our variable, the algebraic expression for this math phrase is simply*

**m****5**. It means

*m***5**times the unknown number

*.*

**m****Example 6**: Write an algebraic expression for the math phrase ” twice a number”.

** Solution**: The word “twice” means we are going to double something. In this case, we want to double an unknown value or quantity. Let the letter

*be the unknown number, when we double it we get the algebraic expression*

**d****2**.

*d***Example 7**: Write an algebraic expression for the math phrase ” the quotient of a number and 7″.

** Solution**: The keyword “quotient” means that we are performing the operation of division. We will divide an unknown number by

**7**. Choosing the letter

*as our variable, the math phrase above can be expressed as the algebraic expression below.*

**w****Example 8**: Write an algebraic expression for the math phrase ” the ratio of 10 and a number”.

** Solution**: Similarly, the word “ratio” means division. The order here is very important. The first quantity is the number

**10**and the second quantity is the unknown number. That means

**10**is divided by an unknown number. Let

*be the unknown number, the algebraic expression for the math phrase above can be written as*

**c**