Algebraic expressions have variables, constants, and mathematical operators that represent all the values of a given variable. They are commonly found to be represented using words.
For example, in the expression ‘k more than 12’ the keyword is ‘more than,’ which means addition.
Representing the above expression mathematically, we get
k + 12
Thus, what we do here is replace the words (or phrases) with operators.
In the same way, while writing or translating the algebraic expressions into verbal phrases, we replace the operators with the words.
For example, in the expression 7 – p, the operator is ‘-’ (subtraction).
Now, writing the above expression into words, we get,
‘7 is subtracted by p.’
With a Single Operator
Some common words used to write an algebraic expression with their corresponding mathematical operators are given.
Writing Algebraic Expressions
Now, the question is, why should we write an algebraic expression mathematically if we can write it in words?
It is because the mathematical form is simpler and easier to understand and comprehend.
For instance, ‘Twice as much of an unknown amount’ represents 2x.
Here, we can take any unknown amount (x) and twice it, which means twice of x (2x).
Let us consider another example: ‘p decreased by 12.’
Here, the phrase ‘decreased by’ stands for subtraction.
Thus, the algebraic expression is p – 12.
We must be careful about the order in which the numbers or the variables are written.
In the above example, ‘p decreased by 12’ denotes to start with p, then decreases by 12 (i.e., p – 12).
On the other hand, 12 – p represents ‘12 decreased by p.’
With Multiple Operators
Sometimes, we use multiple mathematical operators to write an algebraic expression. In such a case, we use the order of operations (PEMDAS) to solve the problem further.
Let us translate the following word expressions into the algebraic form.
a) Seven times the sum of 4 and n
Here, the keywords are ‘times’ and ‘the sum of.’
Thus, there will be multiplication and addition in the expression.
Since we multiply 7 by a sum, we must add 4 and n first and then multiply the sum by 7.
Thus, the sum is (4 + n), and then, multiplying by 7, we get 7(4 + n).
b) Double the difference of w and 8
Here, the keywords are ‘double’ and ‘the difference of.’
Thus, there will be multiplication and subtraction in the expression.
After doubling the difference, we must multiply the difference by 2.
Thus, (w – 8) is multiplied by 2 = 2(w – 8).
From Word Problems
Many of these phrases or keywords are familiar to us in our surroundings. Representing real-world situations in algebraic form is also very helpful.
Let us interpret and write an algebraic expression from a word problem:
‘Mrs. Johnson likes to divide her class into groups of 5.’
To represent all the students in her class mathematically,
Let ‘f’ be the number of groups in Mrs. Johnson’s class.
Then, ${5\cdot f}$ or 5f represents ‘f groups of 5 students’.
Solved Examples
Henry bought c pounds of cashews at the grocery store for $4 per pound. How much will they cost?
Solution:
Since the cost of cashews per pound is dollar 4. The cost for c pounds of cashews is dollar ${4\cdot c}$. Thus, Henry bought c pounds of cashews at dollar 4c.
Ms. Jones put ‘p’ pencils on each of her 20 students’ desks. Choose the expression that shows how many pencils she put on the desks.
Solution:
Ms. Jones put p pencil on each of her students’ desks. Thus, for 20 students, she put ${20\cdot p}$ or 20p pencils on the desks.
Jackson spent 7 of his ‘d’ dollars on a sandwich for lunch. Choose the expression that shows how many dollars Jackson has left.
Solution:
Jackson had d dollars, and he spent 7 dollars for lunch. Thus, Jackson has left with d-7 dollars.
Gloria has dimes and quarters in her pocket. The number of dimes is six less than seven times the number of quarters. Let x represent the number of quarters she has. Write an expression for the number of dimes.
Solution:
Let x be the number of quarters. Thus, the number of dimes = 6 less than 7 times the number of quarters = 6 less than 7x = 7x – 6. The number of dimes is 7x – 6.
