# 3.1: Mathematical Expressions

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall the definition of a *variable* presented in Section 1.6.

A variable is a symbol (usually a letter) that stands for a value that may vary.

Let’s add the definition of a *mathematical expression*.

When we combine numbers and variables in a valid way, using operations such as addition, subtraction, multiplication, division, exponentiation, and other operations and functions as yet unlearned, the resulting combination of mathematical symbols is called a *mathematical expression*.

Thus,

2*a*, *x* + 5, and *y*^{2},

being formed by a combination of numbers, variables, and mathematical operators, are valid mathematical expressions. A mathematical expression must be *well-formed*. For example,

2 + ÷5*x*

is *not a valid expression* because there is no term following the plus sign (it is not valid to write +÷ with nothing between these operators). Similarly,

2 + 3(2

is not well-formed because parentheses are not balanced.

## Translating Words into Mathematical Expressions

In this section we turn our attention to translating word phrases into mathematical expressions. We begin with phrases that translate into *sums*. There is a wide variety of word phrases that translate into sums. Some common examples are given in Table \(\PageIndex{1a}\), though the list is far from complete. In like manner, a number of phrases that translate into differences are shown in Table \(\PageIndex{1b}\).

Phrase | Translates to: | Phrase | Translates to: |
---|---|---|---|

sum of x and 12 |
x + 12 |
difference of x and 12 |
x − 12 |

4 greater than b |
b + 4 |
4 less than b |
b − 4 |

6 more than y |
y + 6 |
7 subtracted from y |
y − 7 |

44 plus r |
44 + r |
44 minus r |
44 − r |

3 larger than z |
z + 3 |
3 smaller than z |
z − 3 |

a) Phrases that are sums | b) Phrases that are differences |

Let’s look at some examples, some of which translate into expressions involving sums, and some of which translate into expressions involving differences.

Translate the following phrases into mathematical expressions:

- "12 larger than
*x,*" - "11 less than
*y*," and - "
*r*decreased by 9."

###### Solution

Here are the translations.

- “12 larger than x” becomes
*x*+ 12. - “11 less than y” becomes
*y*− 11. - “r decreased by 9” becomes
*r*− 9.

Translate the following phrases into mathematical expressions:

- "13 more than
*x*" and - "12 fewer than
*y*".

**Answer**-
(a)

*x*+ 13 and(b)

*y*− 12

Let *W* represent the width of the rectangle. The length of a rectangle is 4 feet longer than its width. Express the length of the rectangle in terms of its width *W*.

###### Solution

We know that the width of the rectangle is *W*. Because the length of the rectangle is 4 feet longer that the width, we must add 4 to the width to find the length.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length} & \text{is} & \colorbox{cyan}{4} & \text{more than} & \colorbox{cyan}{the width} \\ \text{Length} & = & 4 & + & W \end{array}\nonumber \]

Thus, the length of the rectangle, in terms of its width *W*, is 4 + *W*.

The width of a rectangle is 5 inches shorter than its length *L*. Express the width of the rectangle in terms of its length *L*.

**Answer**-
*L*− 5

A string measures 15 inches is cut into two pieces. Let *x* represent the length of one of the resulting pieces. Express the length of the second piece in terms of the length *x* of the first piece.

###### Solution

The string has original length 15 inches. It is cut into two pieces and the first piece has length *x*. To find the length of the second piece, we must subtract the length of the first piece from the total length.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length of the second piece} & \text{is} & \colorbox{cyan}{Total length} & \text{minus} & \colorbox{cyan}{the length of the first piece} \\ \text{Length of the second piece} & = & 15 & - & x \end{array}\nonumber \]

Thus, the length of the second piece, in terms of the length x of the first piece, Answer: 12 + *x* is 15 − *x*.

A string is cut into two pieces, the first of which measures 12 inches. Express the total length of the string as a function of *x*, where *x* represents the length of the second piece of string.

**Answer**-
12 +

*x*

There is also a wide variety of phrases that translate into products. Some examples are shown in Table 3.2(a), though again the list is far from complete. In like manner, a number of phrases translate into quotients, as shown in Table 3.2(b).

Phrase | Translates to: | Phrase | Translates to: |
---|---|---|---|

product of x and 12 |
12x |
quotient of x and 12 |
x/12 |

4 times b |
4b |
4 divided by b |
4/b |

twice r |
2r |
the ratio of 44 to r |
44/r |

a) Phrases that are products. | b) Phrases that are differences. |

Let’s look at some examples, some of which translate into expressions involving products, and some of which translate into expressions involving quotients.

Translate the following phrases into mathematical expressions: (a) “11 times *x*,” (b) “quotient of *y* and 4,” and (c) “twice *a*.”

###### Solution

Here are the translations. a) “11 times *x*” becomes 11*x*. b) “quotient of *y* and 4” becomes *y*/4, or equivalently, \(\frac{y}{4}\). c) “twice *a*” becomes 2*a*.

Translate into mathematical symbols: (a) “the product of 5 and *x*” and (b) “12 divided by *y*.”

**Answer**-
(a) 5

*x*and (b) 12/*y*.

A plumber has a pipe of unknown length *x*. He cuts it into 4 equal pieces. Find the length of each piece in terms of the unknown length *x*.

###### Solution

The total length is unknown and equal to *x*. The plumber divides it into 4 equal pieces. To find the length of each pieces, we must divide the total length by 4.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length of each piece} & \text{is} & \colorbox{cyan}{Total length} & \text{divided by} & \colorbox{cyan}{4} \\ \text{Length of each piece} & = & x & \div & 4 \end{array}\nonumber \]

Thus, the length of each piece, in terms of the unknown length *x*, is *x*/4, or equivalently, \(\frac{x}{4}\).

A carpenter cuts a board of unknown length *L* into three equal pieces. Express the length of each piece in terms of *L*.

**Answer**-
L/3

Mary invests *A* dollars in a savings account paying 2% interest per year. She invests five times this amount in a certificate of deposit paying 5% per year. How much does she invest in the certificate of deposit, in terms of the amount A in the savings account?

###### Solution

The amount in the savings account is A dollars. She invests five times this amount in a certificate of deposit.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Amount in CD} & \text{is} & \colorbox{cyan}{5} & \text{times} & \colorbox{cyan}{Amount in savings} \\ \text{Amount in CD} & = & 5 & \cdot & A \end{array}\nonumber \]

Thus, the amount invested in the certificate of deposit, in terms of the amount A in the savings account, is 5*A*.

David invests *K* dollars in a savings account paying 3% per year. He invests half this amount in a mutual fund paying 4% per year. Express the amount invested in the mutual fund in terms of *K*, the amount invested in the savings account.

**Answer**-
\(\frac{1}{2}K\)

## Combinations

Some phrases require combinations of the mathematical operations employed in previous examples.

Let the first number equal *x*. The second number is 3 more than twice the first number. Express the second number in terms of the first number *x*.

###### Solution

The first number is *x*. The second number is 3 more than twice the first number.

\[ \begin{aligned} \colorbox{cyan}{Second number} & \text{is} & \colorbox{cyan}{3} & \text{more than} & \colorbox{cyan}{twice the first number} \\ \text{Second number} & = & 3 & + & 2x \end{aligned}\nonumber \]

Therefore, the second number, in terms of the first number *x*, is 3 + 2*x*.

A second number is 4 less than 3 times a first number. Express the second number in terms of the first number *y*.

**Answer**-
3

*y*− 4

The length of a rectangle is *L*. The width is 15 feet less than 3 times the length. What is the width of the rectangle in terms of the length *L*?

###### Solution

The length of the rectangle is *L*. The width is 15 feet less than 3 times the length.

\[ \begin{aligned} \colorbox{cyan}{Width} & \text{is} & \colorbox{cyan}{3 times the length} & \text{less} & \colorbox{cyan}{15} \\ \text{Width} & = & 3L & - & 15 \end{aligned}\nonumber \]

Therefore, the width, in terms of the length *L*, is 3*L* − 15.

The width of a rectangle is *W*. The length is 7 inches longer than twice the width. Express the length of the rectangle in terms of its length *L*.

**Answer**-
2

*W*+ 7

## Exercises

In Exercises 1-20, translate the phrase into a mathematical expression involving the given variable.

1. “8 times the width n ”

2. “2 times the length z ”

3. “6 times the sum of the number n and 3”

4. “10 times the sum of the number n and 8”

5. “the demand b quadrupled”

6. “the supply y quadrupled”

7. “the speed y decreased by 33”

8. “the speed u decreased by 30”

9. “10 times the width n ”

10. “10 times the length z ”

11. “9 times the sum of the number z and 2”

12. “14 times the sum of the number n and 10”

13. “the supply y doubled”

14. “the demand n quadrupled”

15. “13 more than 15 times the number p ”

16. “14 less than 5 times the number y ”

17. “4 less than 11 times the number x ”

18. “13 less than 5 times the number p ”

19. “the speed u decreased by 10”

20. “the speed w increased by 32”

21. Representing Numbers. Suppose n represents a whole number.

i) What does n + 1 represent?

ii) What does n + 2 represent?

iii) What does n − 1 represent?

22. Suppose 2n represents an even whole number. How could we represent the next even number after 2n?

23. Suppose 2n + 1 represents an odd whole number. How could we represent the next odd number after 2n + 1?

24. There are b bags of mulch produced each month. How many bags of mulch are produced each year?

25. Steve sells twice as many products as Mike. Choose a variable and write an expression for each man’s sales.

26. Find a mathematical expression to represent the values.

i) How many quarters are in d dollars?

ii) How many minutes are in h hours?

iii) How many hours are in d days?

iv) How many days are in y years?

v) How many months are in y years?

vi) How many inches are in f feet?

vii) How many feet are in y yards?

Answers

1. 8n

3. 6(n + 3)

5. 4b

7. y − 33

9. 10n

11. 9(z + 2)

13. 2y

15. 15p + 13 17.

11x − 4

19. u − 10

21.

i) n+1 represents the next whole number after n.

ii) n+2 represents the next whole number after n + 1, or, two whole numbers after n.

iii) n − 1 represents the whole number before n.

23. 2n + 3

25. Let Mike sell p products. Then Steve sells 2p products.