Word problems

Math Word Problem Solver

Paste the paragraph and get the variable definitions, the equation each sentence produces, and an answer checked against the original wording.

Solve a word problem problem now

First solution free

Almost nobody who struggles with word problems struggles with the algebra in them. The equations are usually simpler than the ones in the same week's exercise set. What breaks is the step before: deciding what the letter stands for, and turning each English sentence into a statement about that letter. So the translation is written out here as its own stage, with the sentence and its equation side by side.

Paste the whole question. Extra information is part of the problem, and the numbers you decide to ignore should be a conclusion rather than an assumption. For a deeper treatment of the method, the guide on turning words into equations works through the same framework at length.

What this solver handles

Age problems, including ones set partly in the past or future.
Distance, rate, and time — meeting, overtaking, round trips, and travel against a current.
Mixture and concentration, where the conserved quantity is the amount of solute rather than the volume.
Work rate problems, including ones with a drain working against a fill.
Percentage change, discount, markup, tax, and simple or compound interest.
Consecutive integer and digit puzzles, using the standard phrase mappings.
Geometry in prose: a perimeter or area constraint that becomes an equation in one unknown.
Two-unknown problems — coins, tickets, two-item baskets — set up as a system of equations.
Ratio, proportion, and direct or inverse variation.

Include the final sentence of the question even when you are sure you know what it asks. It is the sentence that decides whether the answer is one person's age or the sum of two, and it is the one most often left out of a paste. Keep the units in as well: “in 3 years” and “in 3 months” produce different equations from otherwise identical text, and the difference is invisible once the numbers have been stripped out.

The phrases that carry the operators

Most of the translation is a small vocabulary applied consistently. The two rows worth staring at are the second and fourth, because one of them reverses the order in which the words arrive and the other does not.

English	Algebra
the sum of a number and 7	n + 7
8 less than a number	n − 8
twice a number, decreased by 3	2n − 3
the quotient of a number and 4	n ÷ 4
three consecutive integers	n, n + 1, n + 2
is, was, gives, results in	=
15 per cent of a number	0.15n
how many more A than B	A − B

“Eight less than a number” is $n - 8$ , even though 8 is spoken first. “The quotient of a number and 4” is $n \div 4$ , in the order given. Subtraction and division are the two operations that care about order, so those are the two phrases worth slowing down for.

Three problems, worked

Worked example

Maria is 4 years older than twice her brother’s age. In 3 years their ages sum to 43.

1
Define the variable for the quantity everything else is described in terms of. Maria's age depends on her brother's, so the brother gets the letter.
$b = brother’s age now, in years$
2
Translate the first sentence. 'Twice her brother's age' is 2b, and '4 years older' adds 4 to that.
$M = 2 b + 4$
3
Translate the second sentence. In three years, every person is three years older — the phrase applies to both of them, which is the step most often missed.
$(b + 3) + (2 b + 4 + 3) = 43$
4
Collect like terms on the left.
$3 b + 10 = 43$
5
Solve the linear equation.
$3 b = 33 ⟹ b = 11$
6
Return to the question, which asked for both ages. Check against both sentences: 26 is four more than twice 11, and in three years 14 plus 29 is 43.
$M = 2 (11) + 4 = 26$

Answer

Brother 11, Maria 26

Worked example

A train leaves at 60 mph. Two hours later a second leaves the same station at 75 mph.

1
Choose the variable carefully. Let t be the travel time of the second train, because that is what the question asks for.
$t = hours the second train travels$
2
The first train left two hours earlier, so it has been travelling longer by exactly that amount.
$t + 2 = hours the first train travels$
3
Catching up means both have covered the same distance from the station. Distance is rate times time for each train.
$60 (t + 2) = 75 t$
4
Expand and collect the t terms.
$60 t + 120 = 75 t ⟹ 15 t = 120$
5
Solve, then translate back into the units of the question.
$t = 8 hours after the second train departs$
6
Check both distances agree: the second covers 75 × 8 = 600 miles, and the first covers 60 × 10 = 600 miles. Equal distances confirm the setup, not just the arithmetic.

Answer

8 hours, 600 miles from the station

Worked example

One pipe fills a tank in 6 h, a second in 4 h, and an open drain empties it in 12 h.

1
Times do not add; rates do. Convert each duration into a fraction of the tank per hour.
$\frac{1}{6}, \frac{1}{4}, \frac{1}{12}$
2
The drain works against the pipes, so its rate is subtracted. Sign is the whole difficulty in this problem type.
$\frac{1}{6} + \frac{1}{4} - \frac{1}{12}$
3
Use a common denominator of 12 to combine them.
$\frac{2}{12} + \frac{3}{12} - \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$
4
A combined rate of one third of a tank per hour means the time to fill one whole tank is the reciprocal.
$T = \frac{1}{1/3} = 3 hours$
5
Sanity check the size of the answer: with the drain shut the faster pipe alone would take 4 hours, so any answer above 4 would have been wrong before the arithmetic was checked.

Answer

3 hours

The check for a word problem is not substitution into your own equation — that only confirms you solved what you wrote down. Read the answer back into the original sentences instead. Maria at 26 with a brother of 11 satisfies both “four years older than twice” and the sum of 43 in three years. If an answer satisfies one sentence and not the other, the fault is in the translation rather than the algebra, and rechecking the algebra will find nothing.

Magnitude is the second filter and it costs nothing. An age cannot be negative, a volume of one component cannot exceed the total, a discounted price cannot exceed the original, and two pipes working together cannot take longer than the faster pipe alone. Deciding what range the answer must fall in before computing turns a wrong answer into something you catch rather than something you hand in.

Translate one yourself

Cover the explanations and try each line first. The discipline being practised is not solving — it is writing the sentence in symbols before deciding what to do with it.

A rectangle's length is 3 cm more than twice its width. Its perimeter is 42 cm.

0 of 5 steps

Try it yourself first — you'll remember it longer.

Where students go wrong

Never defining the variable

If $x$ is not written down as a specific quantity with units, it drifts. Halfway through a distance problem it silently becomes a time, and the equation stops describing anything. One line — let t be the time in hours for the second train — prevents most of the errors below it.

Reversing subtraction or division

“Five less than a number” is $n - 5$ , not $5 - n$ . Addition and multiplication forgive the reversal because they commute; subtraction and division do not. Read those two phrases twice before writing them down.

Mixing units inside one equation

Minutes with hours, centimetres with metres, or grams with kilograms will produce an answer that looks fine and is off by a factor of 60 or 1000. Convert everything into one system before the first equation is written, and keep the units attached to the numbers as you go.

Solving for x and stopping there

The question often wants something built from $x$ : the older brother's age, the total cost, the number of the other kind of coin. Finish by rereading the final sentence of the problem and confirming that the thing you wrote down is the thing it asked for.

The translation breaks on one line

One of these lines is wrong. Click it.

Three of those four happen before any algebra begins and the fourth happens after it ends. None of them is in the middle, which is exactly why rechecking your working never turns them up. The remedy in every case is to slow down at the one line where English becomes symbols, and to write the variable definition with its units on the page rather than holding it in your head. The reading strategy behind that is set out in how to read a word problem without panicking.

Formulas worth knowing cold

Word problem reference

Tap any formula with a derivation to see where it comes from.

Distance, rate, time

Mixture and work

Percentage and interest

Practice

Four to try

Answers are checked here — nothing is sent anywhere.

1
$Twice a number decreased by 7 is 19. Find the number.$
2
$Two numbers sum to 47 and differ by 9.$
3
$How many litres of 10% acid must join 10 L of 40% to give 25% ?$
4
$A paints a room in 5 h, B in 20 h. Together?$

Frequently asked questions

Is the word problem solver free?

The first solution is free and needs no account. A free account then gives three solutions a day, and Gauth Plus is $11.99 a month with a three-day free trial if you want the cap removed.

Does it show the steps or just the answer?

It shows the translation as well as the algebra: what each letter stands for, which sentence produced which equation, and a final check that reads the answer back against the original wording.

Can I paste the whole paragraph?

Paste it exactly as printed, including the units and any information that looks irrelevant. Deciding what matters is part of the problem, and trimming the text yourself can remove the constraint that made it solvable.

Does it work for problems with two unknowns?

Yes. Two unknowns need two independent statements, and the solution identifies both sentences before setting up the system, so you can see where each equation came from.

Will it explain why my own equation was wrong?

Paste your setup and ask. Most word problem errors happen in the translation rather than the algebra, so having the faulty equation named is usually more useful than a correct solution.

Can it handle problems in another language?

Type the problem in the language it was set in and the solution follows in the same language. The algebra is identical; only the phrase-to-operator mapping changes.

Keep going

Turning words into equationsThe five-step framework and the full phrase-to-operator table.Substitution, elimination, or graphing?What to do once the paragraph has produced two equations.Algebra solverFor the equation itself, once the translation is settled.Equation solver with stepsEvery operation applied to both sides, one per line.System of equations solverTwo unknowns, two sentences, one worked system.