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Word Problem Calculator

Turn any worded question into an equation and solve it, one step at a time
Maria is 3 times as old as Ben. In 8 years she will be twice as old as Ben. How old are they now?
A train leaves at 60 mph. Two hours later a second train leaves the same station at 80 mph. When does it catch up?
A jacket is reduced by 30% and then 8% tax is added, giving $75.60. What was the original price?
The mean of five test scores is 82. Four of them are 78, 85, 90 and 74. Find the fifth.

Translating Words into Equations

Word problems are rarely hard algebra; the difficulty is the translation. Work in a fixed order:

  1. Name the unknown. Write a full sentence: "let bb be Ben's age in years now." A vague variable is where most errors start.
  2. Write what you are told, one relationship per line, in the same units.
  3. Form the equation from the sentence that contains the comparison — usually the one with is, equals, total or same as.
  4. Solve it.
  5. Answer the question asked, with units — often that is not the variable you solved for.
  6. Check the numbers back in the original wording, not in your own equation.

The usual translations

WordsAlgebra
is, gives, results in==
of (with a fraction or percent)×\times
more than, increased by++
less than xxx−□x - \square (note the order)
twice, three times as many2x2x, 3x3x
per, each, for everya rate to multiply by
in tt yearsadd tt to every age

Patterns Worth Recognising

Most school word problems are one of a handful of templates.

  • Age: define ages now, then add the same offset to everyone for a future or past comparison.
  • Distance–rate–time: d=rtd = rt. Set up whichever quantity is shared — equal distances for a catch-up, a total distance for a two-leg trip, a total time for a round trip.
  • Mixture and concentration: track the pure amount, not the volume. 0.30(x)+0.10(20−x)0.30(x) + 0.10(20 - x) is litres of solute, and that is what balances.
  • Work rate: add rates, not times. Together, 1a+1b=1t\frac{1}{a} + \frac{1}{b} = \frac{1}{t}.
  • Percent change: a 30%30\% discount multiplies by 0.700.70; 8%8\% tax multiplies by 1.081.08. Chained changes multiply, they do not add.
  • Statistics: the mean gives you a total. If xˉ=∑xn\bar{x} = \frac{\sum x}{n} then ∑x=nxˉ\sum x = n\bar{x}, which converts a "missing value" question into one-step algebra.

The assumption people forget: the equation only encodes what the words actually say. If a problem says "the second train leaves two hours later", the two trains do not travel for the same time, and any equation that gives them the same tt is solving a different problem.

Common Mistakes to Avoid

  • Reversing "less than". "Seven less than xx" is x−7x - 7, never 7−x7 - x.
  • Mismatched units. Minutes with hours, cents with dollars, cm with m — convert before forming the equation.
  • Applying a change to only one side of a comparison. In ten years, both people age ten years.
  • Adding percentages of different bases. A 30%30\% cut then an 8%8\% tax is 0.70×1.08=0.7560.70 \times 1.08 = 0.756, not a 22%22\% cut.
  • Answering the variable instead of the question. If xx is the number of adult tickets but the question asks for the revenue, one more step remains.
  • Skipping the check. Substituting back into the sentence catches translation errors that substituting into your own equation never will.

Examples

Step 1: Let bb be Ben's age now in years; then Maria's age now is 3b3b
Step 2: In 8 years: Ben is b+8b + 8 and Maria is 3b+83b + 8
Step 3: The comparison sentence gives 3b+8=2(b+8)3b + 8 = 2(b + 8)
Step 4: Expand: 3b+8=2b+163b + 8 = 2b + 16
Step 5: Subtract 2b2b and 88 from both sides: b=8b = 8, so Maria is 3(8)=243(8) = 24
Step 6: Check against the wording: in 8 years Ben is 16 and Maria is 32, and 32=2×1632 = 2 \times 16
Answer: Ben is 8 years old and Maria is 24 years old

Step 1: Let tt be the time in hours after the second train departs
Step 2: The first train has been travelling t+2t + 2 hours, so it has gone 60(t+2)60(t + 2) miles
Step 3: The second train has gone 80t80t miles
Step 4: They meet where the distances are equal: 60(t+2)=80t60(t + 2) = 80t
Step 5: Expand and solve: 60t+120=80t⇒120=20t⇒t=660t + 120 = 80t \Rightarrow 120 = 20t \Rightarrow t = 6 hours
Step 6: Distance: 80×6=48080 \times 6 = 480 miles; check the first train: 60×8=48060 \times 8 = 480 miles
Answer: The second train catches up 6 hours after it departs, 480 miles from the station

Step 1: The mean is xˉ=∑xn\bar{x} = \frac{\sum x}{n}, so the total is ∑x=nxˉ\sum x = n\bar{x}
Step 2: ∑x=5×82=410\sum x = 5 \times 82 = 410
Step 3: Add the four known scores: 78+85+90+74=32778 + 85 + 90 + 74 = 327
Step 4: The fifth score is 410−327=83410 - 327 = 83
Step 5: Check: (78+85+90+74+83)/5=410/5=82(78 + 85 + 90 + 74 + 83)/5 = 410/5 = 82
Answer: The fifth score is 83

Frequently Asked Questions

Define the unknown in a full sentence with its units — "let b be Ben's age in years now" — then write each fact as its own equation and look for the sentence containing a comparison such as "is", "equals" or "in total". That sentence becomes the equation you solve.

"Seven less than x" means x − 7, with the order reversed from the way it is read. This is the single most common translation error; "seven minus x" is the phrase that means 7 − x.

Use d = rt and find the quantity the two travellers share. For a catch-up problem the distances are equal, so express each distance as rate × time and set them equal — remembering that a head start means the two times differ by the head start.

Yes. Statistics word problems usually reduce to one relationship — for the mean, Σx = n·x̄, which turns a missing-value question into one-step algebra. Probability, standard deviation and confidence-interval wording are handled the same way: identify the formula the sentence describes, then substitute.

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