# LESSON: Translating English Phrases and Sentences into Mathematical Phrases and Sentences

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## Introduction

If we know how to do Algebra, we will be able to use math concepts to solve a wide range of problems in real life. To start, you should be able to turn simple English or spoken sentences and phrases into mathematical sentences and phrases, and vice versa. With this information, it will be easy to figure out how to solve the problem using math.

Knowing the symbols for known and unknown quantities, as well as the symbols for operations and relationships, will make it easier to turn spoken sentences into mathematical sentences.

## Mathematical phrases

Math statements that are written in English are called verbal or English phrases. You will be able to write different math expressions once you know what these words mean.

Mathematical phrases, on the other hand, are statements made with symbols like letters, numbers, and operations.

There are a lot of Mathematical symbols but we're just going to look at the most important.

## Addition  $+$

This is the symbol of addition. It represents the addition of one number to another. We can read this symbol of operations as Addition, Plus, increased by, added to, the sum of, more.

## Subtraction $-$

This is the subtraction symbol. It represents the subtraction of one number from another.. We can read this symbol as Subtraction, minus, decreased by, subtracted from, less than, diminished by.

## Multiplication $x$

The symbol multiplication represents the number of times that you need to add a number. We can also use the following symbol to represent multiplication $.$ (dot symbol) and $( )$ (parenthesis). We can read this as multiplication, times, multiplied by, product of.

## Division $\div$ , $/$

This is the division symbol. It represents the total number of times you need to share equally. It can also read as division, divided by, ratio of, quotient of

## Greater than and less than $<,>$

These are the greater than and less then symbols. We use them to compare quantities.

## Greater than or equal to and less than  or equal to $\le , \ge$

The greater than or equal to symbol that is used to represent inequality in math. It tells us that the given variable is either greater than or equal to a particular value. while  the 'less than or equal to' sign is just the opposite of this sign. Less than or equal to means, the amount is equal to or less than the maximum limit.

## Not equal $\ne$

The not equal sign is an unequal synonym. The not equal symbol is a sign of "inequality". It is meant to show a comparison between the two quantities which are unequal hence, representing inequality among them.

The use of the symbols of operation together with symbols such as letters and numbers will enable us to translate verbal phrases into mathematical phrases. It is important for us to be able to associate various key words with their related arithmetic operation.

Aside from the above mentioned symbols, We can also find a clue on the four basic operations, there are keywords that can signal which operation will be used.

The following chart lists some of those keywords and phrases.

Some words and phrases can signal more than one operation.

For example, the word each might mean multiplication, as it did in the raffle ticket example at the beginning of the lesson. However, if we were told that a class reads $250$ books and we are looking for how many books each student read, each would signal division.

Example 1: the total of three and four

The total of three and four  would be an addition expression: $3 + 4$. The keyword $total$ tells us to add three and four.

Example 2: the difference between ten and six

The phrase the difference between ten and six is a subtraction phrase, because of the keyword difference. The difference between ten and six is $10 – 6$.

Sometimes, the numbers given in a phrase appear in the opposite order that we will use them when we form our number sentence. This can happen with subtraction and division phrases, where the order of the numbers is important.

Example 3: five times the difference between eleven and four

The phrase five times the difference between eleven and four means that 4 must be subtracted from $11$ before multiplication occurs. We must use parentheses to show that subtraction should be performed before multiplication: $5(11 – 4)$.

The order of the addends in an addition sentence, or the order of the factors in a multiplication sentence, does not matter. The phrase one fewer than five is a subtraction phrase, but be careful. One fewer than five is $5 – 1$, not $1 – 5$. Some phrases combine more than one operation: ten more than five minus three can be written as $10 + 5 – 3$ or  $(5 – 3 + 10)$. As we learned in our previous posts, the order of operations is important, and it is just as important when forming number sentences from phrases. Seven less than eight times negative two is $(8)(–2) – 7$. We must show that $8$ and $–2$ are multiplied, and that $7$ is subtracted from that product. Although multiplication comes before subtraction in the order of operations, a phrase might be written in such a way that subtraction must be performed first.

Lets have a practice! You are going to write each phrase as numerical expression.

1. five increased by two

The keyword increased signals addition. Five increased by two is 5 + 2.

2. the quotient of sixteen and four

The keyword quotient signals division. The quotient of sixteen and four is 16 ÷ 4.

3. four less than the sum of two and nine

The keywords less than signal subtraction and the keyword sum signals addition. The sum of two and nine is 2 + 9, and four less than that sum is (2 + 9) – 4.

4. negative twelve times three fewer than fifteen

The keywords fewer than signal subtraction and the keyword times signals multiplication. Three fewer than fifteen is 15 – 3. Negative twelve times that quantity is –12(15 – 3).

5. the difference between the total of one and seven and the product of six and three

The keyword difference signals subtraction, but before we can subtract, we have to translate the total of one and seven and the product of six and three into numbers and operations. The keyword total signals addition. The total of one and seven is (1 + 7). The keyword product signals multiplication. The product of six and three is (6)(3). Now we are ready to write the difference between the two quantities. The difference between the total of one and seven and the product of six and three is (1 + 7) – (6)(3).