Learn to write algebraic expressions in and out of word problems.

Why do we have math if we can describe things in words?

Sometimes in math we describe an expression with a phrase. For example, the phrase
"22 more than 55"
can be written as the expression
2+52 + 5.
Similarly, when we describe an expression in words that includes a variable, we're describing an algebraic expression (an expression with a variable).
For example,
"33 more than xx"
can be written as the algebraic expression
x+3x+ 3.
But why? Why use math if we can describe things in words? One of the many reasons is that math is more precise and easier to work with than words are. This is a question you should keep thinking about as we dig deeper into algebra.

Different words for addition, subtraction, multiplication, and division

Here is a table that summarizes common words for each operation:
OperationWordsExample algebraic expression
Additionplus, sum, more than, increased byx+3x + 3
Subtractionsubtracted, minus, difference, less than, decreased byp6p - 6
Multiplicationtimes, product8k8k
Divisiondivided, quotienta÷9a \div 9
For example, the word "product" tells us to use multiplication. So, the phrase
"the product of 88 and kk"
can be written as
8k8k.

Let's take a look at a trickier example

Write an expression for "mm decreased by 77".
Notice that the phrase "decreased by" tells us to use subtraction.
So the expression is m7m - 7.
The word "decreased" is used, which means we should subtract, but we still need to be careful about which order to put the numbers in.
For example, "mm decreased by 77" tells us to start at mm and decrease by 77 which can be written as the expression m7m - 7.
On the other hand, the expression 7m7 - m is "77 decreased by mm."

Let's try some practice problems!

Basic expressions

Write an expression for "yy more than 77".
Write an expression for "66 minus bb".
Write an expression for "xx divided by 99".
Write an expression for "22 multiplied by tt".
Write an expression for "the sum of aa and 44".

More complicated expressions

Write an expression for "88 less than the product of 77 and xx".
Write an expression for "the sum of 66 and the product of 33 and dd".

Word problems

Daniel makes 100100 dollars each week. He worked for xx weeks this summer.
How much money did Daniel make this summer?
Write your answer as an expression.
Let's see what happens as the number of weeks increases:
Number of weeksMoney Daniel made
1\blueD{1}1001=100100 \cdot \blueD{1} = 100
2\blueD{2}1002=200100 \cdot \blueD{2} = 200
3\blueD{3}1003=300100 \cdot \blueD{3} = 300
x\blueD{x}100x=100x100 \cdot \blueD{x} = 100x
The answer: 100x100x
Sebastian has 1212 more trophies than Megan. Megan has tt trophies.
How many trophies does Sebastian have?
Write your answer as an expression.
Let's see what happens as the amount of trophies Megan has increases:
Megan's trophy countSebastian's trophy count
10\blueD{10}10+12=22\blueD{10}+12 = 22
20\blueD{20}20+12=32\blueD{20}+12 = 32
30\blueD{30}30+12=42\blueD{30}+12 = 42
t\blueD{t}t+12\blueD{t}+12
The answer: t+12t+12