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Patterns and Expressions

Write variable expressions to represent word problems.

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Patterns and Expressions

Jeremy read that degrees Celsius converted to degrees Fahrenheit is "the sum of 32 and \begin{align*}\frac{9}{5}\end{align*}95 times the temperature in degrees Celsius." However, he's not sure how to convert this into an algebraic expression. What do you think an equivalent algebraic expression would be? 

Writing Algebraic Expressions

In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables, expressions can be written to describe a pattern. Recall that an algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations. We can describe patterns using phrases as well, and we want to be able to translate these phrases into algebraic expressions.

Consider a theme park charging an admission of $28 per person. A rule can be written to describe the relationship between the amount of money taken at the ticket booth and the number of people entering the park. In words, the relationship can be stated as “The money taken in dollars is (equals) twenty-eight times the number of people who enter the park.”

The English phrase above can be translated (written in another language) into an algebraic expression. Using mathematical verbs and nouns, any phrase can be written as an algebraic expression.

Let's write an algebraic expression for the each of the following phrases:

  1. The product of \begin{align*}c\end{align*}c and 4.

The verb is product, meaning “to multiply.” Therefore, the phrase is asking for the answer found by multiplying \begin{align*}c\end{align*}c and 4. The nouns are the number 4 and the variable \begin{align*}c\end{align*}c. The expression becomes \begin{align*}4 \times c,\end{align*}4×c,which may also be written as \begin{align*}4(c), \text{ or }4c.\end{align*}4(c), or 4c. 

  1. The phrase about the theme park from above: The money taken in dollars is (equals) twenty-eight times the number of people who enter the park.

An appropriate variable to describe the number of people could be \begin{align*}p\end{align*}p. Rewriting the English phrase into a mathematical phrase, it becomes \begin{align*}28 \times p\end{align*}28×p.

  1. 5 less than 2 times a number.

Some phrases are harder to translate than others. The word "less" lets you know that you are going to take away, or subtract, a number. Many students will want to turn this expression into \begin{align*}5-2n\end{align*}52n, but this is not what the phrase is telling us. Whatever the value of "2 times a number "or \begin{align*}2n\end{align*}2n, we want to write an expression that shows we have 5 less than that. That means that we need to subtract 5 from \begin{align*}2n\end{align*}2n. The correct answer is \begin{align*} 2n-5\end{align*}2n5.


Example 1

Earlier, you were asked told that the degrees Fahrenheit temperature is "the sum of 32 and \begin{align*}\frac{9}{5}\end{align*}95 times the temperature in degrees Celsius". What is an equivalent algebraic expression?

We can use \begin{align*}c\end{align*}c to represent the degrees Celsius. The word "sum" indicates addition and the word "times" indicates multiplication. The multiplication between \begin{align*}c\end{align*}c and \begin{align*}\frac{9}{5}\end{align*}95 occurs first and then 32 is added to the result. Therefore the expression to represent the phrase is:

\begin{align*}32 + \frac{9}{5}c\end{align*}32+95c

Example 2

A student organization sells shirts to raise money for events and activities. The shirts are printed with the organization's logo and the total costs are $100 plus $7 for each shirt. The students sell the shirts for $15 each. Write an expression for the cost and an expression for the revenue (money earned).

We can use \begin{align*}x\end{align*}x to represent the number of shirts. For the cost, we have a fixed $100 charge and then $7 times the number of shirts printed. This can be expressed as \begin{align*} 100+7x\end{align*}100+7x. For the revenue, we have $15 times the number of shirts sold, or \begin{align*} 15x\end{align*}15x.


For exercises 1 – 15, translate the English phrase into an algebraic expression. For the exercises without a stated variable, choose a letter to represent the unknown quantity.

  1. Sixteen more than a number
  2. The quotient of \begin{align*}h\end{align*}h and 8
  3. Forty-two less than \begin{align*}y\end{align*}y
  4. The product of \begin{align*}k\end{align*}k and three
  5. The sum of \begin{align*}g\end{align*}g and \begin{align*}-7\end{align*}7
  6. \begin{align*}r\end{align*}r minus 5.8
  7. 6 more than 5 times a number
  8. 6 divided by a number minus 12
  9. A number divided by \begin{align*}-11\end{align*}11
  10. 27 less than a number times four
  11. The quotient of 9.6 and \begin{align*}m\end{align*}m
  12. 2 less than 10 times a number
  13. The quotient of \begin{align*}d\end{align*}d and five times \begin{align*}s\end{align*}s
  14. 35 less than \begin{align*}x\end{align*}x
  15. The product of 6, \begin{align*}-9\end{align*}9, and \begin{align*}u\end{align*}u

In exercises 16 – 24, write an English phrase for each algebraic expression

  1. \begin{align*}J - 9\end{align*}J9
  2. \begin{align*}\frac{n}{14}\end{align*}n14
  3. \begin{align*}17-a\end{align*}17a
  4. \begin{align*}3l-16\end{align*}3l16
  5. \begin{align*}\frac{1}{2} (h)(b)\end{align*}12(h)(b)
  6. \begin{align*}\frac{b}{3} + \frac{z}{2}\end{align*}b3+z2
  7. \begin{align*}4.7-2f\end{align*}4.72f
  8. \begin{align*}5.8 + k\end{align*}5.8+k
  9. \begin{align*}2l+2w\end{align*}2l+2w

In exercises 25 – 28, define a variable to represent the unknown quantity and write an expression to describe the situation.

  1. The unit cost represents the quotient of the total cost and number of items purchased. Write an expression to represent the unit cost of the following: The total cost is $14.50 for \begin{align*}n\end{align*}n objects.
  2. The area of a square is the side length squared.
  3. The total length of ribbon needed to make dance outfits is 15 times the number of outfits.
  4. What is the remaining amount of chocolate squares if you started with 16 and have eaten some?

Use your sense of variables and operations to answer the following questions.

  1. Describe a real-world situation that can be represented by \begin{align*}h + 9\end{align*}h+9.
  2. What is the difference between \begin{align*}\frac{7}{m}\end{align*}7m and \begin{align*}\frac{m}{7}\end{align*}m7?

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.6. 

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algebraic expression An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.
\therefore The symbol \therefore means "therefore" or "because of this."
Algebraic Equation An algebraic equation contains numbers, variables, operations, and an equals sign.
domain The domain of a function is the set of x-values for which the function is defined.
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
horizontal axis The horizontal axis is also referred to as the x-axis of a coordinate graph. By convention, we graph the input variable on the x-axis.
Range The range of a function is the set of y values for which the function is defined.
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.
vertical axis The vertical axis is also referred to as the y-axis of a coordinate graph. By convention, we graph the output variable on the y-axis.

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