Why is "5 less than x" written as x-5 instead of 5-x?

Because the phrase describes starting at x and taking 5 away. The quantity named right after "than" is the starting amount. Test it with plain numbers: 5 less than 12 is 7, and 12-5=7, while 5-12=-7. The same reversal happens with "subtracted from": "5 subtracted from x" is also x-5. Note that "the difference of 5 and x" does not reverse — it is 5-x.

When do I need parentheses in a translated expression?

Whenever an operation applies to an entire quantity rather than a single number. The tip-off is a grouping phrase such as "the sum of", "the difference of", or "the quantity" appearing after another operation word, as in "twice the sum of n and 4", which is 2(n+4). Commas in the sentence often mark the same boundary. Fraction bars group automatically, so x+7/2 needs no extra parentheses in the numerator.

What is the difference between an expression and an equation?

An expression such as 9h+25 is a phrase — it has a value once you know h, but it makes no claim. An equation such as 9h+25=79 contains an equals sign and states that two quantities are equal, so it can be solved. This lesson builds expressions; words like "is", "equals", "results in", and "totals" are the signals that an equation is coming, which is the focus of later lessons.

Does it matter which letter I pick for the variable?

Mathematically, no — 9h+25 and 9x+25 describe the same rule. Practically, choosing a letter that matches the quantity, like h for hours or c for classes, makes your work far easier to reread and check. What does matter is writing a definition sentence that states the quantity and its unit, such as "let h = the number of hours the canoe is rented", so the expression can be interpreted correctly later.

ALG1-1.4

Translating Words into Algebraic Expressions

Learn to define a variable and turn word phrases into algebraic expressions, including tricky order-sensitive wording like "less than", "subtracted from", and "twice the sum of".

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Translating Words into Algebraic Expressions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Translating Words into Algebraic Expressions free

What this lesson covers

Algebra is often called a language, and this lesson is where you start translating. Every word problem you meet for the rest of the course begins with the same two moves: decide what the unknown quantity is, name it with a variable, and then rewrite the English sentence as a symbol sentence.

Most of the translation is straightforward — "more than" means add, "of" usually means multiply. But a handful of phrases reverse the order of the numbers, and those phrases cause more mistakes than anything else in Unit 1. By the end of this lesson you will be able to write expressions like

3(n+4)-5

from a sentence, explain exactly why "7 less than

x

" is

x-7

and not

7-x

, and set up expressions for real situations such as flat fees plus hourly rates.

Start by Defining the Variable

An algebraic expression is a combination of numbers, variables, and operations with no equals sign. Before you write one, you have to say what your letter stands for — and "let

x

be the number" is rarely specific enough.

A good variable definition names the quantity and its unit. Compare these two:

Weak: Let

t

= time.

Strong: Let

t

= the number of hours the bike is rented.

The strong version tells you, three steps later, whether

60t

means dollars, hours, or miles. When a problem contains two related unknowns, define one of them and write the other in terms of it. If Jordan is 5 years older than Casey, do not invent two letters. Write: let

c

= Casey's age in years; then Jordan's age is

c+5

.

A quick way to check your definition is to substitute a number into it. If

c=12

, does

c+5=17

actually describe Jordan? Yes. If you had written

c-5

, plugging in would give 7, which would make Jordan younger — the check catches the error immediately.

One more habit worth building now: the variable represents a number, never an object. "Let

a

= apples" leads to nonsense like

3a

meaning three apples instead of three times the count. Write "let

a

= the number of apples" and the arithmetic stays honest. This distinction matters enormously in the next unit when you build equations from these expressions.

Operation Keywords and What They Signal

Most translation runs on a small vocabulary. Learn these groups rather than memorizing dozens of individual sentences.

Operation	Words that signal it	Phrase	Expression
Addition	sum, plus, more than, increased by, total, gained	the sum of $n$ and 8	$n+8$
Subtraction	difference, minus, less than, decreased by, fewer, subtracted from	4 decreased by $n$	$4-n$
Multiplication	product, times, twice, triple, of, per, each	the product of 6 and $n$	$6n$
Division	quotient, per, divided by, split evenly, ratio	the quotient of $n$ and 3	$\frac{n}{3}$

Notice that "per" appears in two rows. In "12 dollars per ticket,

t

tickets" it multiplies:

12t

. In "miles per gallon" it divides. Context, not the word alone, decides.

Addition and multiplication are commutative, so

n+8

and

8+n

are both correct translations of "the sum of

n

and 8". Standard form puts the variable term first, but neither version is wrong mathematically.

Subtraction and division are not commutative.

n-8

and

8-n

are different expressions, and

\frac{n}{3}

is not

\frac{3}{n}

. That is why the next section exists.

Also watch for words that describe the whole structure rather than an operation: "a number" introduces the variable, "result" or "total" often marks where an equals sign will go later, and "consecutive" signals a pattern like

n

n+1

n+2

The Order-Sensitive Phrases

Three phrases decide the order of the numbers for you, and two of them write the numbers in reverse of the order you read them. These are the ones to slow down for.

"Less than" — "7 less than

x

" means you start at

x

and take 7 away:

x-7

. The number that comes right after "than" is the starting amount. Test it with numbers: 7 less than 20 is 13, and

20-7=13

. Writing

7-x

would give

7-20=-13

, which is not what the English says.

"Subtracted from" — "9 subtracted from

m

" is

m-9

. The quantity after "from" is what you subtract out of. Contrast this with "

m

subtracted by 9", which is nonstandard phrasing, and with "the difference of

m

and 9", which keeps the reading order:

m-9

.

"Quotient of" — "the quotient of

k

and 5" is

\frac{k}{5}

; the first quantity named is the numerator, the dividend. But "5 divided into

k

" also gives

\frac{k}{5}

, while "

k

divided into 5" gives

\frac{5}{k}

. The word "into" flips things, so translate it carefully or rewrite it as "divided by" first.

Phrase	Correct	Common wrong answer
6 less than $y$	$y-6$	$6-y$
$y$ less 6	$y-6$	—
6 subtracted from $y$	$y-6$	$6-y$
the difference of 6 and $y$	$6-y$	$y-6$
the quotient of $y$ and 6	$\frac{y}{6}$	$\frac{6}{y}$

The reliable defense is the substitution check: replace the variable with an easy number, compute what the English sentence describes, then compute your expression. If they disagree, you flipped it.

Grouped Phrases and Parentheses

Some phrases apply an operation to an entire quantity rather than a single number, and those need parentheses.

The signal is a phrase like "the sum of", "the difference of", or "the quantity" appearing after another operation word. In "twice the sum of

n

and 4", the doubling applies to the whole sum, so the expression is

2(n+4)

. Without parentheses,

2n+4

doubles only

n

— a different expression. Check with

n=3

: twice the sum of 3 and 4 is

2 \cdot 7 = 14

, while

2n+4

gives 10.

Commas in the original sentence often mark the grouping. "Three times the sum of a number and 5" is

3(x+5)

, but "three times a number, plus 5" is

3x+5

. Read punctuation as if it were parentheses.

Division creates grouping automatically because the fraction bar acts as a grouping symbol. "The quotient of the sum of

x

and 7, and 2" is

\frac{x+7}{2}

— no parentheses needed in the numerator, since the bar already holds it together.

Now stack the ideas. "Five less than twice the sum of a number and 4" builds from the inside out: the sum is

x+4

; twice that sum is

2(x+4)

; five less than that means subtract 5 from the whole thing, giving

2(x+4)-5

.

Students most often go wrong by dropping the parentheses or by attaching the "5 less" to the wrong piece. Building outward one layer at a time, and writing each layer down, prevents both errors. In the next lesson you will use the distributive property to rewrite

2(x+4)-5

2x+3

— but only if the parentheses were there to begin with.

Translating Real-World Scenarios

Word problems bundle several phrases into one situation. The structure is almost always the same: a fixed amount that does not change, plus a rate that gets multiplied by a variable quantity.

A gym charges a 40 dollar membership fee plus 15 dollars for each fitness class. Let

c

= the number of classes attended. The fee happens once, so it is a constant: 40. The class charge happens

c

times, so it is

15c

. Total cost in dollars:

15c+40

.

Identifying which number is the rate and which is the constant is the main skill. Words like "each", "per", "every", and "for every hour" attach to the variable. Words like "one-time", "flat", "initial", "starting", and "deposit" mark the constant.

Situation	Variable	Expression
8 dollars per shirt, 6 dollar shipping	$s$ = number of shirts	$8s+6$
Tank starts with 50 gallons, drains 3 gallons per minute	$m$ = minutes	$50-3m$
A number of students split a 90 dollar bill evenly	$p$ = number of people	$\frac{90}{p}$
Perimeter of a rectangle 4 cm longer than it is wide	$w$ = width in cm	$2w+2(w+4)$

Always attach units in your head. In

15c+40

c

is a count of classes,

15c

is dollars, and 40 is dollars — the pieces being added match, which confirms the setup makes sense. If you ever find yourself adding hours to dollars, the translation went wrong somewhere. That unit check will save you repeatedly when these expressions become equations in Unit 2.

Key terms

Variable.: A letter that represents an unknown or changing number. It stands for a quantity, never for an object or a label.
Algebraic expression.: A combination of numbers, variables, and operation symbols with no equals sign, such as $3x-7$ .
Constant.: A term with a fixed value that does not change as the variable changes, like the 40 in $15c+40$ .
Coefficient.: The number multiplied by a variable in a term. In $15c$ , the coefficient is 15 and it usually comes from a rate such as cost per item.
Term.: A single number, variable, or product of numbers and variables, separated from other terms by plus or minus signs.
Quotient.: The result of division. In "the quotient of $a$ and $b$ ", $a$ is the numerator, giving $\frac{a}{b}$ .
Grouping phrase.: Wording such as "the sum of" or "the quantity" that follows another operation and requires parentheses, as in $2(n+4)$ .
Defining a variable.: Writing a sentence stating exactly what quantity the letter represents, including its unit, before building the expression.

Worked example

A canoe rental shop charges a 25 dollar launch fee plus 9 dollars for every hour the canoe is out. (a) Define a variable and write an expression for the total cost in dollars. (b) Separately, translate this phrase into an algebraic expression: "six less than twice the sum of a number and 3". (c) Check part (b) by substituting the number 5 for the variable.

Part (a). First find the quantity that varies. The launch fee happens once no matter what, but the 9 dollar charge repeats for each hour, so hours is the unknown.

Define it precisely: let

h

= the number of hours the canoe is rented.

The hourly charge is 9 dollars for each of

h

hours, which is

9h

dollars. The launch fee is a constant 25. Add them:

9h+25

Unit check:

9h

is dollars and 25 is dollars, so adding them is legitimate. Test it — 3 hours should cost

9(3)+25=52

dollars, which matches counting 27 dollars of hourly charges plus the 25 dollar fee.

Part (b). Build from the inside out. Let

x

= the number.

Innermost layer, "the sum of a number and 3":

x+3

.

Next layer, "twice the sum": the doubling applies to the entire sum, so parentheses are required:

2(x+3)

.

Outermost layer, "six less than" that: "less than" reverses the reading order, so 6 is subtracted from the quantity already built, not the other way around:

2(x+3)-6

Part (c). Substitute

x=5

and compare with the English.

By the words: the sum of 5 and 3 is 8; twice 8 is 16; six less than 16 is 10.

By the expression:

2(5+3)-6=2(8)-6=16-6=10

.

The two agree, so the translation is correct. Notice what the common wrong versions would have given:

2x+3-6

yields 7, and

6-2(x+3)

yields

-10

. Neither matches the sentence, which is exactly how the substitution check exposes a flipped or ungrouped translation.

Practice questions

Which expression represents "eight less than the quotient of a number and 4"?

$8-\frac{n}{4}$
$\frac{n}{4}-8$
$\frac{4}{n}-8$
$\frac{n-8}{4}$

Answer: $\frac{n}{4}-8$

Handle the quotient first: "the quotient of a number and 4" names

n

first, so

n

is the numerator, giving

\frac{n}{4}

. Then "eight less than" that quantity subtracts 8 from it, reversing the reading order, so the 8 goes after the minus sign:

\frac{n}{4}-8

. Check with

n=40

: the quotient is 10, and eight less than 10 is 2, matching

\frac{40}{4}-8=2

. The choice

8-\frac{n}{4}

flips the subtraction,

\frac{4}{n}-8

flips the quotient, and

\frac{n-8}{4}

divides after subtracting, which the sentence does not say.

A phone repair shop charges a flat 30 dollar diagnostic fee plus 45 dollars per hour of labor. Define a variable, write an expression for the total charge in dollars, and use it to find the cost of a job that takes 2.5 hours. Explain how you know which number becomes the coefficient.

Answer: Let $h$ = the number of hours of labor. The total charge in dollars is $45h+30$ . For $h=2.5$ : $45(2.5)+30=112.5+30=142.50$ dollars. The 45 becomes the coefficient because it is attached to the words "per hour", meaning it repeats once for every hour, while the 30 is charged only one time and is therefore the constant.

The key move is sorting the two numbers by whether they repeat. Signal words such as "per", "each", and "every" mark a rate, which multiplies the variable; words such as "flat", "one-time", and "initial" mark a constant, which is simply added. A useful confirmation is the unit check:

45h

has units of dollars per hour times hours, which is dollars, so it can be added to the 30 dollars. Reversing the roles to get

30h+45

would wrongly claim the diagnostic fee is charged every hour.

Explain, using a numerical check, why "three times the sum of a number and 7" is not the same as "three times a number, plus 7".

Answer: The first phrase is $3(x+7)$ and the second is $3x+7$ . Substituting $x=2$ : the first gives $3(2+7)=3(9)=27$ , while the second gives $3(2)+7=6+7=13$ . Because the results differ, the expressions are different; the parentheses are what tell the 3 to multiply the entire sum rather than just the variable.

The phrase "the sum of" following "three times" bundles

x

and 7 into a single quantity, so parentheses are required to preserve that grouping in symbols. In the second phrase the comma separates the multiplication from the addition, so only

x

is tripled. Substituting a specific number is the fastest way to prove two expressions differ — one counterexample is enough. In the next lesson, the distributive property will show that

3(x+7)

can be rewritten as

3x+21

, which makes the gap between the two expressions even clearer.

FAQ

Why is "5 less than x" written as $x-5$ instead of $5-x$ ?: Because the phrase describes starting at $x$ and taking 5 away. The quantity named right after "than" is the starting amount. Test it with plain numbers: 5 less than 12 is 7, and $12-5=7$ , while $5-12=-7$ . The same reversal happens with "subtracted from": "5 subtracted from $x$ " is also $x-5$ . Note that "the difference of 5 and $x$ " does not reverse — it is $5-x$ .
When do I need parentheses in a translated expression?: Whenever an operation applies to an entire quantity rather than a single number. The tip-off is a grouping phrase such as "the sum of", "the difference of", or "the quantity" appearing after another operation word, as in "twice the sum of $n$ and 4", which is $2(n+4)$ . Commas in the sentence often mark the same boundary. Fraction bars group automatically, so $\frac{x+7}{2}$ needs no extra parentheses in the numerator.
What is the difference between an expression and an equation?: An expression such as $9h+25$ is a phrase — it has a value once you know $h$ , but it makes no claim. An equation such as $9h+25=79$ contains an equals sign and states that two quantities are equal, so it can be solved. This lesson builds expressions; words like "is", "equals", "results in", and "totals" are the signals that an equation is coming, which is the focus of later lessons.
Does it matter which letter I pick for the variable?: Mathematically, no — $9h+25$ and $9x+25$ describe the same rule. Practically, choosing a letter that matches the quantity, like $h$ for hours or $c$ for classes, makes your work far easier to reread and check. What does matter is writing a definition sentence that states the quantity and its unit, such as "let $h$ = the number of hours the canoe is rented", so the expression can be interpreted correctly later.

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