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# Mathematical Expressions

## Write variable expressions to represent word problems

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Practice Mathematical Expressions
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Translate Verbal Phrases into Variable Expressions

### [Figure1] License: CC BY-NC 3.0

Twins Larry and Terry want to buy their dad a great gift for his upcoming birthday. Both boys have been saving all of their dimes and quarters for several months. “I count 928 dimes,” Larry announces. “Great! I count 515 quarters,” Terry says. “But how do we go from knowing the number of coins to the amount of money?” How can Larry and Terry use a mathematical expression to figure out how much money they have in total?

In this concept, you will learn to change words into variable expressions.

### Translating Phrases into Expressions

Many problems can be solved using mathematical methods. In order to do this, you must be able to translate words into mathematical expressions. More often than not, the words can be expressed as a variable expression which can be evaluated to solve a problem.

You know the symbols+,,×,÷\begin{align*}+,-,\times , \div\end{align*}, represent the four most commonly used mathematical operations. The operation of multiplication can be represented by the symbol(×)\begin{align*}(\times)\end{align*}or by#( )\begin{align*}\# ( \ )\end{align*}or( )( )\begin{align*}( \ ) ( \ )\end{align*}. Division can be represented by the symbol(÷)\begin{align*}( \div) \end{align*}or##\begin{align*}\frac{\#}{\#}\end{align*}.

In addition to the words add, subtract, multiply and divide, there is a variety of other words that also correspond to these mathematical operations.

Let’s start by creating a list of words that could translate into each of the mathematical operations.

 Addition Subtraction Multiplication Division Add Subtract Multiply Divide Increased by Decreased by Product Quotient Plus Minus Times Shared Sum Difference Twice Split between Total Reduced by Of Divided by More Less than

Let’s look at an example of translating words into a variable expression.

Remember that a variable expression has one or more variables (letters) that represent an unknown quantity.

Write the following as a mathematical expression with one variable:

A number increased by four.

First, name the variable. Let ‘n\begin{align*}n\end{align*}’ represent the number.

Next, represent the mathematical operation by its symbol.+\begin{align*}+\end{align*}

Then, insert the constant after the symbol.4\begin{align*}4\end{align*}

\begin{align*}\underbrace{ \text{A} \ \text{number} } _ {\color{blue}n} \ \underbrace{ \text{increased} \ \text{by} }_{\color{red}+} \ \underbrace{ \text{four}}_{\color{blue}4}\end{align*}The variable expression is \begin{align*}n+4\end{align*}.

Let’s look at one more example.

Write the following as a mathematical expression with two variables:

Twice the length less three times the width.

First, name the variables. Let ‘\begin{align*}l\end{align*}’ represent length and let  ‘\begin{align*}w\end{align*}’ represent width.

Next, represent any operations performed on the variables.\begin{align*}2l\end{align*}and \begin{align*}3w\end{align*}

Then, represent the mathematical operation by its symbol.\begin{align*}-\end{align*}

\begin{align*}\underbrace{ \text{Twice the length} }_{\color{blue}2l} \ \underbrace{ \text{less} }_{\color{red}-} \ \underbrace{ \text{three times the width} }_{\color{blue}3w} \ \end{align*}

The variable expression is\begin{align*}2l-3w\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Larry and Terry and all their dimes and quarters.

Larry and Terry need to write a variable expression to show the value of ‘\begin{align*}d\end{align*}’ dimes and ‘\begin{align*}q\end{align*}’ quarters.

First, name the variables. Let ‘\begin{align*}d\end{align*}’ be the number of dimes. Let ‘\begin{align*}q\end{align*}’ be the number of quarters.

Next, express the value of the coins as decimals. Ten cents is\begin{align*}0.10\end{align*}and twenty-five cents is \begin{align*}0.25\end{align*}.

Next, represent any operation performed on the variables.\begin{align*}0.10d\end{align*} and \begin{align*}0.25q\end{align*}

Then, represent the mathematical operation by its symbol.\begin{align*} +\end{align*}

The variable expression is\begin{align*}0.10d+0.25q\end{align*}

How can Larry and Terry use this variable expression to figure out how much money they have?

They have counted the coins and they know they have 928 dimes and 515 quarters.

They have to evaluate the variable expression\begin{align*} 0.10d+0.25q \end{align*} when\begin{align*}d=928\end{align*} and \begin{align*}q=515\end{align*}.

First, substitute the values\begin{align*}d=928\end{align*} and \begin{align*}q=515\end{align*}into the expression.

\begin{align*}0.10(928)+0.25(515)\end{align*}

Next, multiply to clear the parenthesis:
\begin{align*}0.10(928)=92.80 \ \text{and} \ 0.25(515)=128.75\end{align*}

\begin{align*}92.80+128.75=221.55\end{align*}

The answer is\begin{align*}221.55\end{align*}

Larry and Terry can spend\begin{align*}221.55\end{align*}on their dad’s gift.

#### Example 2

Write the following as a mathematical expression with one variable:

One-half of Adam’s age six years ago.

First, name the variable. Let ‘\begin{align*}x\end{align*}’ represent Adam’s age.

Next, represent any operation performed on the variable.\begin{align*}-6\end{align*}

Then, represent the mathematical operation by its symbol.\begin{align*}\frac{1}{2}( \ )\end{align*}

\begin{align*}\underbrace{\text{One-half of }}_ {\color{blue}\frac{1}{2}} \ \underbrace{\text{Adam's age six years ago }}_ {\color{red}{\color{red}(}{\color{blue}x}-{\color{blue}6})}\end{align*}

The variable expression is\begin{align*}\frac{1}{2}(x-6)\end{align*}.

#### Example 3

Write the following as a mathematical expression with two variables:

Four times the number of dimes less twice the number of quarters.

First, name the variables. Let ‘\begin{align*}d\end{align*}’ represent the number of dimes. Let ‘\begin{align*}q\end{align*}’ represent the number of quarters.

Next, represent any operation performed on the variables.

\begin{align*}4d\end{align*}and\begin{align*}2q\end{align*}

Then represent the mathematical operation by its symbol.\begin{align*}–\end{align*}

\begin{align*}\underbrace{\text{Four times the number of dimes}}_ {\color{blue}4d} \ \underbrace{\text{less}}_ {\color{red}-} \ \underbrace{\text{twice the number of quarters}}_ {\color{blue}2q}\end{align*}

The answer is\begin{align*}4d-2q\end{align*}.

#### Example 4

Write the following as a mathematical expression with two variables:

The total tax from the first purchase at 5% and the second purchase at 9%.

First, name the variables. Let ‘\begin{align*}a\end{align*}’ represent the first purchase. Let ‘\begin{align*}b\end{align*}’ represent the second purchase.

Next, represent any operation performed on the variables.\begin{align*}+5 \% a\end{align*}and \begin{align*}9 \% b\end{align*}.

Then, represent the mathematical operation by its symbol.\begin{align*}+\end{align*}

\begin{align*}\text{The} \ \underbrace{\text{total}}_ {+} \ \underbrace{\text{tax form the first purchase at} \ 5 \%}_{ 5 \% a} \ \text{and} \ \underbrace{\text{the second purchase at}\ 9 \%}_ {9 \% b}\end{align*}

The answer is\begin{align*}+5 \% a\end{align*}and \begin{align*}9 \% b\end{align*}.

This answer is incorrect. You must multiply by the percent expressed as a decimal. The total is the sum of the taxes on each of the purchases.

\begin{align*}5 \%=\frac{5}{100}=0.05 \\ 9 \%=\frac{9}{100}=0.09\end{align*}

\begin{align*}\text{The} \ \underbrace{\text{total}} \ \underbrace{\text{tax form the first purchase at} \ 5 \%}_ {\color{blue}0.05 \% a} \ \underbrace{\text{and}}_{\color{red}+} \ \underbrace{\text{the second purchase at } \ 9 \%}_ {\color{blue}0.09 \% b} \end{align*}

The answer is\begin{align*}0.05a+0.09b \end{align*}.

### Review

Write a variable expression to represent each of the following:

1. The sum of a number and twelve.
2. The difference between a number and eight.
3. Three times a number
4. A number squared plus five
5. A number divided by two plus seven
6. Four times the quantity of a number plus six
7. A number times two divided by four
8. A number times six plus the same number times two
9. A number squared plus seven take a way four
10. A number divided by three plus twelve
11. A number times five and another number times six
12. Sixteen less than a number times negative four
13. A number times eight divided by two
14. A number divided by six and another number times negative five
15. A number divided by four plus another number divided by sixteen

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Color Highlighted Text Notes

### Vocabulary Language: English

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

Verbal Expression

A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.