SolveMathAI

Word problem solver

The hard part is the set-up. That is the part this shows you.

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  • Arithmetic to calculus

Word problems fail at the translation, almost never at the algebra. The reliable method is to name the unknown in a sentence, write down what the problem tells you about it, and only then reach for the equation. Every example below shows that set-up line before any solving starts.

Turning words into an equation

Six steps, in this order. Steps one to three are where the marks are.

  1. Say what you are looking for, in words

    Write "let x be the number of litres of 20% solution". A labelled unknown with units prevents solving for the wrong thing.

  2. Write down every quantity given

    Put them in a small table when there are several. Rate problems have three columns: rate, time, distance.

  3. Find the sentence that balances

    Something equals something: total acid before equals total acid after; distance out equals distance back; work done adds to one whole job.

  4. Turn it into symbols

    Now, and only now, write the equation. The translation is the work; what follows is routine.

  5. Solve, keeping the units attached

    Units that do not cancel into the answer’s units are a sign the equation is wrong.

  6. Answer the question that was asked

    Re-read it. If it asked for the sale price, do not stop at the discount. If it asked for both numbers, give both.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

A train travels 180 miles in 2.5 hours. What is its average speed in miles per hour?

speed=distancetime\text{speed} = \frac{\text{distance}}{\text{time}}
  1. Name the unknown
    let s=average speed in mph\text{let } s = \text{average speed in mph}
  2. Use the relationship
    s=180 miles2.5 hourss = \frac{180 \ \text{miles}}{2.5 \ \text{hours}}
  3. Divide
    s=72s = 72
Answer
72 mph72 \ \text{mph}

How many liters of 20% acid solution must be added to 10 liters of 50% acid solution to get a 30% solution?

0.20x+0.50(10)=0.30(x+10)0.20x + 0.50(10) = 0.30(x + 10)
  1. Name the unknown
    let x=litres of the 20% solution\text{let } x = \text{litres of the }20\%\text{ solution}
  2. Acid before equals acid after
    0.20x+0.50(10)=0.30(x+10)0.20x + 0.50(10) = 0.30(x + 10)
  3. Expand both sides
    0.20x+5=0.30x+30.20x + 5 = 0.30x + 3
  4. Collect
    2=0.10x2 = 0.10x
  5. Solve
    x=20x = 20
  6. Check the concentration
    0.2(20)+530=930=30%  ✓\frac{0.2(20) + 5}{30} = \frac{9}{30} = 30\% \;\checkmark
Answer
20 litres20 \ \text{litres}

One painter can paint a room in 6 hours and another in 4 hours. How long do they take together?

16+14=1t\frac{1}{6} + \frac{1}{4} = \frac{1}{t}
  1. Work in jobs per hour, not hours per job
    16 and 14 of the room each hour\frac{1}{6} \ \text{and} \ \frac{1}{4} \ \text{of the room each hour}
  2. Add the rates
    16+14=212+312=512\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}
  3. Time is one job divided by the combined rate
    t=1512=125t = \frac{1}{\frac{5}{12}} = \frac{12}{5}
  4. Convert to hours and minutes
    2.4 hours=2 h 24 min2.4 \ \text{hours} = 2\ \text{h}\ 24\ \text{min}
Answer
2.4 hours=2 h 24 min2.4 \ \text{hours} = 2\ \text{h}\ 24\ \text{min}

Rates add; times do not. Averaging 6 and 4 to get 5 hours is the classic wrong answer, and it is slower than one painter working alone.

You invest $2000 at 5% interest compounded annually. What is it worth after 3 years?

A=P(1+r)tA = P(1 + r)^t
  1. Write the formula
    A=P(1+r)tA = P(1 + r)^t
  2. Substitute
    A=2000(1+0.05)3A = 2000(1 + 0.05)^3
  3. Evaluate the power
    1.053=1.1576251.05^3 = 1.157625
  4. Multiply
    A=2000×1.157625=2315.25A = 2000 \times 1.157625 = 2315.25
Answer
$2315.25\$2315.25

Simple interest would give $2300. The extra $15.25 is the interest earning interest.

Where marks get lost

Solving for the wrong quantity

A problem that asks how much longer, or what the total is, needs one more line after x is found. Re-read the question before writing the final answer.

Averaging rates

Going out at 30 mph and back at 60 mph is not 45 mph overall. Total distance over total time gives 40 mph, because more time is spent at the slower rate.

Mixing percentages and amounts

In a mixture problem the percentages do not add; the amounts of the substance do. The equation balances litres of acid, not percentages.

Questions

How do I solve word problems in math?

Name the unknown, write down what each sentence says as an equation, solve it, then read the answer back into the original wording to check it makes sense.

How do I know what to set x to?

Set x to what the question asks for, when you can. If two quantities are related, set x to the smaller one and write the other in terms of it, which keeps the numbers positive.

How do I solve distance, rate and time problems?

Use distance = rate × time, and build a row for each traveller. What the two rows share, usually a common distance or a common time, becomes the equation.

How do I solve age problems?

Write everyone present age in terms of one letter, then add or subtract the same number of years to every age for the past or future. The relationship in the question becomes the equation.

How do I handle mixture and concentration problems?

Track the amount of the pure substance, not the total volume. Amount equals concentration times volume, and the amounts add up even when the percentages do not.

What if the problem has too many words?

Underline only the numbers and the relationships between them. Most of a word problem is context, and stripping it leaves two or three facts that translate straight into algebra.

Why does my answer come out negative?

Usually the relationship was written the wrong way round, as older-than instead of younger-than. Check the direction of the comparison before changing anything else.

Can it read a problem from a photo?

Yes. Take a picture of the question and it reads the text, shows you what it read so you can correct it, and then solves it step by step.