Problem solving

Word Problem Translation: Words Into Equations

The algebra in a word problem is rarely the hard part. The translation is, and it follows a fixed five-step procedure.

By the Gauth editorial teamHow this content is produced and checkedUpdated 10 September 2026

The idea

A word problem is a translation exercise between two languages. English states relationships loosely and in any order; algebra states them precisely and positionally. Both describe the same situation, and the difficulty is almost never in the algebra — most students who cannot finish a word problem could have solved the equation instantly had someone handed it to them.

The reason translation feels hard is that English hides its structure. In Maya is four years older than twice her brother’s age, the arithmetic runs in a different order from the words: you double the brother’s age first, then add four, even though “four” appears earlier in the sentence. Algebra makes that order explicit as $2 b + 4$ , and it is that explicitness, not the symbols, that makes the equation solvable.

Once you accept that translation is a separate skill from solving, the fix is structural. You do not need to see the whole solution before you start. You need a procedure that turns each sentence into a fragment of algebra, one at a time, and a habit of noticing which single sentence carries the equals sign.

When you need it

Word problems come in a small number of recurring shapes. Recognising the shape tells you what the unknown should be before you have read the numbers:

Comparison. Ages, prices, or counts described relative to each other. Let the unknown be the quantity everything else is compared to, which is usually the smaller or the one mentioned second.
Total. Several quantities summing to a known figure. The sentence containing the total is your equation.
Rate. Distance, work, or flow, all built on $rate \times time = amount$ . Draw a three-column table with one row per mover and the equation appears in a column.
Percent change. A price after a discount or a population after growth. The unknown is nearly always the original value, and the multiplier is $1 \pm r$ .
Geometry in prose. Perimeter, area, or angle relationships. The formula supplies the equation, and the sentences supply the expressions you substitute into it.

When two quantities cannot be reduced to one another in a short phrase, you need two unknowns and two equations — at which point the problem becomes a system to solve by substitution or elimination. The translation procedure below is identical; you simply write two equations instead of one.

The method

Five steps, in this order. The order matters: steps two and three are what make step four possible, and skipping to the equation is the single most common reason a problem stalls.

Read twice, and underline the question. The first pass is for the situation, the second for what is actually being asked. Problems routinely include a number that is never used, and they routinely ask for something other than the quantity you will solve for.
Name the unknown with units. Let b = the brother’s age in years. Not “let $x$ = brother’s age”, and certainly not “let $x$ = brother”. A variable stands for a number, and writing the units down forces you to notice conversions at the start instead of the end.
Express everything else in terms of that unknown. Work through the problem sentence by sentence, converting each into an expression and listing them. Maya’s age becomes $2 b + 4$ . You are building vocabulary, not yet making a claim.
Find the sentence that is an equation. One sentence asserts that two quantities are equal — a total, a perimeter, two costs that match, a distance covered. That sentence becomes the equation, and the expressions from step three fill it in.
Solve, answer in words, and check the story. The algebra is routine. Then convert back: the question asked for an age, not for $b$ . Substitute into the original sentences, and check that the answer is physically sensible.

Most translation errors happen inside step three, and nearly all of them involve the two operations that are not commutative. This table is the part worth knowing by heart:

Phrase	Becomes	Watch for
the sum of a and b	a + b	safe — addition commutes
7 more than n	n + 7	safe in either order
7 less than n	n − 7	reversed from reading order
7 subtracted from n	n − 7	reversed from reading order
the difference of a and b	a − b	follows reading order
twice a number, tripled	2n, 3n	the multiplier attaches to the noun
the quotient of a and b	a ÷ b	a is the numerator
n divided into 12	12 ÷ n	reversed from 'divided by'
15% of x	0.15x	'of' with a percent means multiply
x increased by 15%	1.15x	not 0.15x — the original is still there
is, was, gives, will be	=	this is the sentence that becomes the equation
consecutive integers	n, n + 1, n + 2	—
consecutive even integers	n, n + 2, n + 4	still n + 2, not n + 1

The pattern behind the traps is simple. Since $a + b = b + a$ and $ab = ba$ , getting the order wrong in addition or multiplication costs nothing. Subtraction and division do not commute, so order is the whole meaning:

a - b \neq = b - a and \frac{a}{b} \neq = \frac{b}{a}

That is why the only phrases you have to be careful with — less than, subtracted from, divided into— all involve those two operations. There is no such thing as an “added to” trap.

Solve a word problem problem now

First solution free

Three worked examples

The first is a plain comparison. The second is a percent problem where the obvious translation is wrong. The third is deliberately awkward: two rates, a staggered start, and a question that is not the quantity you solve for.

Worked example

Maya is 4 years older than twice her brother’s age. Their ages total 34. How old is each?

1
The question asks for both ages. Name the unknown as the age everything else is compared to — the brother's, since Maya is described in terms of him.
$Let b = the brother’s age in years$
2
Translate the comparison. 'Twice his age' is 2b, and 'four years older than' that adds 4 afterwards.
$Maya’s age = 2 b + 4$
3
Find the sentence that is an equation. 'Their ages total 34' is the only equality in the problem.
$b + (2 b + 4) = 34$
4
Solve.
$3 b + 4 = 34 ⟹ 3 b = 30 ⟹ b = 10$
5
Answer what was asked, which was both ages, not just b.
$brother = 10, Maya = 2 (10) + 4 = 24$
6
Check against the sentences: 24 is indeed four more than twice 10, and 10 + 24 = 34.
$2 (10) + 4 = 24 ✓ 10 + 24 = 34 ✓$

Answer

The brother is 10 and Maya is 24.

Worked example

A jacket costs £34 after a 15% discount. What was the original price?

1
The unknown is the original price, not the discount. This is the step people skip, and it is why the wrong equation gets written.
$Let p = the original price in pounds$
2
A 15% discount removes 0.15p from p. Write it as a single multiplier rather than two terms — the arithmetic is identical and the structure is clearer.
$p - 0.15 p = 0.85 p$
3
The equation is the sentence stating what the jacket costs now.
$0.85 p = 34$
4
Solve by dividing, not by adding 15% back. Adding 15% to 34 gives 39.10, which is a different and wrong answer.
$p = \frac{34}{0.85} = 40$
5
Check in the story: 15% of 40 is 6, and 40 − 6 = 34.
$0.15 \times 40 = 6, 40 - 6 = 34 ✓$

Answer

The original price was £40.

Worked example

Printer A does 45 pages/min. Printer B does 30 pages/min but starts 4 minutes later. How long from A’s start until 1, 200 pages are printed?

1
The question asks for elapsed time from A's start, so define the unknown that way. Defining it as B's running time is also valid but then the final answer needs an extra step, and that is where the marks go missing.
$Let T = minutes since printer A started$
2
Build the rate table. Each printer contributes rate times its own running time, and B's running time is 4 minutes shorter.
$A: 45 T pages, B: 30 (T - 4) pages$
3
The equation is the sentence about the total output.
$45 T + 30 (T - 4) = 1200$
4
Expand carefully. The 30 multiplies both terms in the bracket, so the constant is −120, not −4.
$45 T + 30 T - 120 = 1200 ⟹ 75 T = 1320$
5
Solve.
$T = \frac{1320}{75} = 17.6 minutes$
6
Check both contributions separately. A runs the full 17.6 minutes; B runs 13.6 minutes.
$45 (17.6) = 792, 30 (13.6) = 408, 792 + 408 = 1200 ✓$
7
Sanity-check the size. Together the printers do 75 pages a minute, so 1,200 pages needs about 16 minutes, plus a little for B's late start. 17.6 fits.
$\frac{1200}{75} = 16 minutes if both ran the whole time$

Answer

17.6 minutes, or 17 minutes 36 seconds, after printer A starts.

Translate one yourself

This sentence contains two of the reversals from the table. Write the equation before you solve it, and take the hints one at a time rather than all three.

Check yourself

Translate into an equation and solve: seven less than three times a number is the same as the number increased by five.

Where people go wrong

Letting the variable stand for a thing instead of a number

“Let $m$ = Maya” leads directly to nonsense like $m + b = 34$ being read as two people summing to 34. A variable is always a number, so the definition must say which number: Maya’s age in years. This sounds pedantic until you meet a problem about coins, where the same letter could plausibly mean the number of coins or their total value, and the two lead to different equations.

Reversing a discount by adding the percentage back

If a price falls 15% to £34, the original is not $34 \times 1.15 = 39.10$ . The 15% was taken from the original, not from the reduced price, so the correct relationship is $0.85 p = 34$ and the answer is £40. The percentages are computed from different bases, which is why the two calculations disagree by more than a rounding error.

Answering with the variable rather than the question

Solving gives $b = 10$ , but the question asked for both ages. A question about consecutive integers may want the largest, and a rate question may want the total time when you solved for one leg of it. Re-read the underlined question before writing the final line, every time.

Distributing a bracket onto only its first term

$30 (T - 4)$ is $30 T - 120$ , not $30 T - 4$ . The translation was correct and the arithmetic destroys it. This is an order-of-operations failure rather than a translation failure, and it is unpicked in the guide to reading an expression the way a grader does.

Practice

Write the equation on paper first, then enter the number only — no units, no variable name.

Four translations

Answers are checked here — nothing is sent anywhere.

1
$Five less than twice a number is 17. Find the number.$
2
$A number decreased by 30% is 84. What was the number?$
3
$Three consecutive even integers sum to 78. What is the largest?$
4
$A rectangle’s length is 3 cm more than its width; the perimeter is 46 cm. Find the width in cm.$

Where this goes next

When one sentence will not reduce two unknowns to one, the same five steps produce two equations instead, and the finish is substitution, elimination, or graphing. Area and projectile problems translate to a squared term, which the quadratic formula handles — and there the plausibility check matters most, since one of the two roots is usually a negative length that has to be discarded. For a longer treatment of reading strategy on genuinely nasty problems, see how to read a word problem without panicking, and paste anything that will not translate into the word problem solver, which shows the English-to-algebra mapping explicitly rather than jumping to an answer.

Frequently asked questions

Where do I start when a word problem looks impossible?

Name the unknown before you understand the whole problem. Most of the paralysis comes from trying to hold the entire situation in your head at once. Writing 'let t = the time in hours until they meet' converts a vague story into a concrete object you can build sentences around, and the rest of the translation usually follows within two lines.

Should I use one variable or two?

Use one if the second quantity can be written in terms of the first in a few words, which is the case for phrases like 'four years older' or 'twice as many'. Use two when the relationship is genuinely symmetric, such as two unknown amounts in a mixture, and expect to need two equations. Two variables with only one equation is the sign that you have missed a sentence.

Why does 'less than' reverse the order?

Because it describes a position relative to something, not an instruction to subtract in reading order. 'Five less than x' names a number five below x, which is x - 5. The same reversal applies to 'fewer than' and 'subtracted from'. Addition has no equivalent trap because it commutes, which is exactly why the error only ever shows up with subtraction and division.

How do I check a word-problem answer?

Substitute back into the original sentences, not into your equation. Your equation may be a faithful translation or it may be the mistake, and checking against it cannot tell the difference. Also test plausibility: a negative length, a person aged 340, or a speed of 4,000 km/h means the translation is wrong even when the algebra is flawless.

Do units really matter if the answer is just a number?

Yes, because carrying them catches errors nothing else will. If one rate is in pages per minute and a time is given in hours, the mismatch is invisible in the numbers and obvious in the units. Writing 'let r = speed in km/h' rather than 'let r = speed' also forces you to convert at translation time instead of discovering the problem three lines from the end.

Keep going

Word problem solverShows the translation from English to algebra before solving.Systems of equationsWhat to do when the problem needs two unknowns.Order of operationsThe arithmetic rules that turn a correct translation into a correct answer.How to read a word problem without panickingSix worked examples of steadily increasing nastiness.