Every word problem in this course, and every one after it, starts the same way: a sentence in English, and a blank page. The algebra you'll learn for solving equations is only useful once a sentence has become an equation in the first place. That step, turning words into symbols, is a skill of its own, separate from the solving that comes after it. This section is entirely about that one step. Nothing here gets solved. The whole point is to get comfortable stopping before solving, at the moment the sentence becomes math.
Why translation is the real skill
Section 05 nailed down what a variable, an expression, and an equation actually are. This section asks a different question: given a sentence, how do you know which one to build? And out of what pieces? "The variable is the unopened box" told you what a letter means once you already have one. It never told you how to decide, from a paragraph of English, which quantity deserves to be the box. That's the missing piece. It's exactly the gap that turns "I understood the lecture" into "I stared at the word problem and didn't know where to start."
Translation is the process of converting a verbal description of a relationship between quantities into a mathematical expression, equation, or inequality. It sits between reading the problem and doing algebra with it. Unlike simplifying an expression or solving for x, it has no fixed procedure to memorize; it requires reading for meaning.
Finding the quantities
Read the whole problem once before writing anything. Then read it again. This time, hunt specifically for the numbers and for the one thing the question is actually asking for. Those are often not the same sentence. A problem can hand you three numbers and only need two of them. The third might be there just to test whether you're building a model, or simply grabbing every digit you see and hoping it combines into something. "How many tickets were sold" and "the theater has 200 seats" can both appear in the same paragraph. The seat count is scene-setting, not an input to the equation.
Before writing any symbol, identify: what is known, what is unknown, what units are involved, and what is actually being asked. A number that appears in the problem but plays no role in the relationship being described is irrelevant to the model; noticing that is part of reading the problem correctly, not a shortcut around it.
Check your understanding (counts toward your progress)
Let n = a number. Write an expression for "5 less than the number." Use n.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
"5 less than the number" takes the number and removes 5 from it: n - 5. The 5 is what's being subtracted, even though it's named first in the sentence — English word order doesn't have to match symbol order here.
Correct answer
Also accepted:
Why
The quantity being taken AWAY is "5" — figure out which symbol that becomes and which side of the minus sign it goes on.
Which expression matches "the quotient of a number n and 6"?
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"Quotient of A and B" means A divided by B, in that order — this answer divides B by A instead.
This is a subtraction, not a quotient — "quotient" always signals division.
This is a product, not a quotient — "quotient" always signals division.
Why
"Quotient of A and B" translates to A/B: the first-named quantity is the numerator, the second is the divisor, so "the quotient of n and 6" is n/6.
"Quotient of A and B" always means A divided by B, first-named on top.
Defining a variable, with units
"Let x = tickets" leaves a real question unanswered: does x count tickets, or price them, or time how long they took to sell? Naming the quantity precisely closes that gap. "Let x = the number of tickets sold" settles it up front, before you're staring at x = 4.5 later and can't remember whether a fraction is even a reasonable answer for what x was counting. Units solve a different kind of ambiguity. "Let t = time" doesn't say whether t=2 means two minutes or two hours, and a distance/rate/time problem falls apart the moment two quantities are secretly measured in different units.
A variable definition names the specific quantity a symbol stands for, including its units where units exist. "Let w = the width" is incomplete if the problem is measured in feet — "let w = the width of the garden, in feet" is not. A variable that names an object ("let g = the garden") rather than a quantity is not usable in an equation at all, since an equation relates numbers, not things.
Two inequality phrases, two pictures
Compare the shaded region for "at least 20" against the one for "no more than 100" — notice which end is shaded and whether the boundary point is filled in.
Loading interactive…
Check your understanding (counts toward your progress)
Which variable definition below is incomplete?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
This one is complete — it names the quantity, the units, and the reference point time is measured from.
This one is complete — the quantity (area) and its units (square feet) are both named.
This one is complete — the quantity (total cost) and its units (dollars) are both named.
Why
"Let t = time" names a quantity but gives no units and no reference point — time since when, measured how? Every other option pairs its quantity with a unit.
A time variable needs more than just "time" — in what unit, and measured from what starting point?
A problem asks about a garden's width. Which is the best variable definition?
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This names the OBJECT (the garden), not the quantity being measured — a variable stands for a number, not a thing.
This names the quantity but leaves out its units — is w in feet, meters, or something else?
This names a unit, not a quantity — there's no number here for w to actually represent.
Why
A complete variable definition names the specific quantity AND its units — "the width of the garden, in feet" tells a reader exactly what w measures and what a value like w = 12 would mean.
A good variable definition answers two questions at once — what is being measured, and in what units?
The phrase table
It's tempting to want a lookup table you can apply mechanically: see "more," write "+." Resist that. Keywords are clues, not a mechanical translation system. They narrow down what a phrase probably means, but the actual sentence still has to be read to confirm it. The table below is exactly that: a set of clues worth recognizing on sight, not a substitute for understanding the sentence they came from.
Common phrases recur often enough to be worth a reference, but a phrase's meaning always comes from its context, never from the words in isolation. The table below is a starting vocabulary, not a substitute for reading the sentence.
| language | mathematical interpretation |
|---|---|
| sum of, total, combined, increased by, more than | addition |
| difference between, decreased by, fewer than, less than | subtraction (watch the order) |
| product of, times, of (with a fraction/percent), twice, double | multiplication |
| quotient of, ratio of A to B, split evenly, per | division (A/B; "per" also signals a rate) |
| is, was, gives, results in, equals | = |
| at least, no less than | \geq |
| at most, no more than | \leq |
| p\% of | multiply by p/100 |
| consecutive integers | n,\ n+1,\ n+2,\ \dots |
Worked example
Building an expression piece by piece
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1
"3 more than twice a number": translate this phrase into an expression.
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2
Identify the variable. Let n = the number.
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3
Translate to algebra. "Twice a number" is 2n. "3 more than" that adds 3 on top: 2n + 3. Notice the phrase table's "more than" clue (addition) confirmed what the sentence already meant; it didn't replace reading it.
Sort each phrase by its operation
Click a phrase, then click the bin it belongs in.
Loading interactive…
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Correct answer
Also accepted:
Why
Why
The order trap
This is the single most common way a translation goes wrong, so it earns its own callout instead of hiding inside the phrase table above. "5 less than x" and "5 is less than x" look almost identical on the page and mean completely different things. The first has no verb "is" in it. It's a recipe, an expression, and it comes out to x - 5: take x, remove 5. The second has "is" sitting right in the middle, turning it into a claim you could check: a statement, 5 < x. One extra word changes the object entirely, from something you compute to something you compare. Whenever "less than" shows up, pause and ask two questions. Is there an "is" anywhere nearby, making it a statement instead of an expression? And if it's an expression, which quantity is actually being taken away? The one named right before "less than" is always the one that disappears. It goes second in the expression, after the minus sign, no matter where it sits in the original sentence.
English word order does not always match symbol order. "A less than B" translates to B - A, not A - B: the quantity named first is the one being subtracted, not the one it's subtracted from. This applies to "less than," "subtracted from," "decreased by" used in reverse, and any phrase where the thing removed is named before the thing it's removed from. Compare this to a genuinely different structure: the word "is" turns a phrase into a statement rather than an expression — "5 is less than x" is the comparison 5 < x, not an arithmetic operation at all.
Worked example
5 less than x, vs. 5 is less than x
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1
Translate both "5 less than x" and "5 is less than x."
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2
Identify the variable. x is already named in both phrases; no new variable is needed here.
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3
Translate to algebra. "5 less than x" has no "is": it's an expression. The 5 is removed from x: x - 5. "5 is less than x" has "is": it's a statement comparing two quantities, 5 < x. Same five words rearranged, same two symbols involved, genuinely different objects. One you could simplify further; the other you could only ever say is true or false.
Check your understanding (counts toward your progress)
Solving a word problem about the number of people at a party gives x = -3. What does this tell you?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
A negative number of people isn't a value that exists in the real situation — the mathematical solution and a valid solution IN CONTEXT aren't automatically the same thing.
There's no justification for rounding to 0 specifically — a negative result here is a signal to check the setup, not a value to patch into something plausible-looking.
Dropping the sign isn't a valid move — it isn't given that the magnitude is right and only the sign is wrong; the whole solution needs re-examining.
Why
A count of people is never negative, so x = -3 is a mathematical solution with no valid meaning in this context — the answer is to go back and check the equation's setup (or reconsider whether the situation is possible as described), not to force -3 into a real-world answer.
Ask whether a negative value is even POSSIBLE for the quantity this variable represents, before reporting any number as the final answer.
Let x = a number. Write an expression for "the difference between x and 5" (in that order). Use x.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
"The difference between A and B" (in that order) means A minus B: x - 5.
Correct answer
Also accepted:
Why
"Difference between A and B" keeps the order it's given in — A first, then minus B.
Let x stand for a number. Translate "5 less than x" into an algebraic expression using x.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
"5 less than x" takes x and removes 5 from it: x - 5. This is an EXPRESSION, with no claim of equality — compare this to "5 is less than x," which is a completely different statement.
Correct answer
Also accepted:
Why
The quantity being subtracted is named first in the sentence, but it still goes second in the expression — after the minus sign.
Translate, don't solve
Every exercise so far in this section stopped at an expression or a statement. None of them asked you to find a number. That's deliberate. Solving is Chapter 3's job, and by then you'll have real tools for it: the same balance-scale moves from Section 05, extended to handle whatever the equation looks like. For now, the skill worth building on its own is stopping at the right spot. Turn the sentence into something correct on the page. Then check, just by looking at it, that it faithfully represents what the sentence said, before a single step of solving happens.
Setting up the correct equation, expression, or inequality is a distinct skill from solving it, and it's worth practicing on its own. A student who can solve any equation handed to them but can't build the equation in the first place has not yet learned to model, only to compute. The questions in this section ask only for the setup.
Where this goes next
Every later chapter
This section has no dramatic real-world payoff of its own. Its payoff is everywhere else. From here forward, nearly every chapter in this course includes problems that start as a sentence: a rectangle's dimensions, a phone plan's cost, an investment's growth, two trains approaching each other. Every one of them uses exactly the moves practiced here: find the quantities, name a variable with its units, and turn the relationship into symbols. Only then does that chapter's specific algebra get used. Skipping this section doesn't mean avoiding translation forever; it means re-deriving it from scratch, under pressure, the first time a word problem that actually matters shows up.
Practice: Translating words into math (5 problems · doesn’t affect your progress)
Let n = a number. Write an expression for "the sum of a number and 10." Use n.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
"Sum of A and B" translates directly to A + B: n + 10.
Correct answer
Also accepted:
Why
A problem asks how long a hike takes. Which is the best variable definition?
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This names the activity, not a quantity — a variable needs to stand for a number.
This names the quantity but not its units — hours, minutes, and days would all fit "time."
This names a unit with no quantity attached to it.
Why
A complete variable definition names the quantity AND its units together — "the time the hike takes, in hours" leaves no ambiguity about what a value like t=2.5 would mean.
Let t = the time the hike takes, in hours.
Let x = a number. Write an expression for "8 less than x." Use x.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
The quantity being taken away (8) is named first in the sentence, but it still goes second in the expression, after the minus sign: x - 8.
Correct answer
Also accepted:
Why
A student translated "4 less than three times a number n" as 4 - 3n. What should it be instead?
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This subtracts 4 from n before tripling, but the original phrase triples the number FIRST, then takes 4 away from that result.
Test it against a number — if n = 10, "4 less than three times 10" should give 30 - 4 = 26, not 4 - 3(10) = -26. The sign comes out backwards.
"Less than" signals subtraction, not addition.
Why
"Less than" reverses word order from symbol order: the amount taken away (4) is named first, but goes second in the expression, after the minus sign — 3n - 4.
3n - 4
A width comes out to w = -6 inches when solving a geometry word problem. What should you conclude?
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A width doesn't have a "direction" to be negative in — this isn't a real physical quantity.
Discarding the sign without knowing WHY it's negative risks reporting a wrong value that happens to look plausible.
A negative root often means the OTHER root (or a re-examined setup) gives the real answer — it's a signal to check, not a dead end.
Why
A negative width has no physical meaning, so w = -6 is a mathematically valid solution to the equation that isn't valid IN CONTEXT — the right move is to check whether another root exists, or whether the equation was set up correctly, not to force the number into an answer.
Something is wrong — a physical width can't be negative, so the setup or an earlier step needs a second look.
What's next
Quantities identified, variables named with units, phrases turned into symbols, and the one trap ("less than") that catches almost everyone at least once: that's the whole translation toolkit. It's the one you'll reach for by name every time a later chapter's word problem starts with a sentence instead of an equation. The next section turns to the notation itself: reading a string like \{x \in \mathbb{R} : x > -2\} fluently, the same way you'd read a sentence out loud.