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Math solver examples by topic

Sample problems for every topic, with two solutions worked out on the page. Open any problem in the solver with one click and compare the explanation levels.

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Topic
Explanation
0 / 300 words

Solution

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At a glance

This page collects 21 sample problems across arithmetic, algebra, geometry, trigonometry, calculus, statistics and word problems, three per topic, plus two solutions worked by hand in the solver's layout. Copy any problem, or press Try it to open it in the solver, then choose Answer + steps or Detailed teaching.

Samples
21 problems in 7 topics
Worked solutions
2, written by hand in the solver's layout
Price
1 credit per problem ($0.08 to $0.10); packs from $4.99
Problem length
Typed text, up to 300 words per problem

What these examples show

  • 01

    Every topic

    Three problems each for arithmetic, algebra, geometry, trigonometry, calculus, statistics and word problems.

  • 02

    Common traps

    Fractions with unlike denominators, stacked discounts, degrees versus radians and sample versus population.

  • 03

    One click to try

    Press Try it to open a problem in the solver, then choose the topic and the explanation level.

How to use an example

  1. 01

    Pick a problem

    Choose one from your topic, or one close to your homework.

  2. 02

    Try both levels

    Solve with Answer + steps, then read the Detailed teaching version of a hard one.

  3. 03

    Change the numbers

    Edit the problem to match your own and solve again; each solution uses 1 credit.

Worked example: a derivative

Written by hand in the Detailed teaching layout, shortened to the main lines:

  • Problem: Find f'(x) for f(x) = x^2 sin(x)
  • Idea: f is a product of two functions, so use the product rule (uv)' = u'v + uv'
  • 1. u = x^2, so u' = 2x
  • 2. v = sin(x), so v' = cos(x)
  • 3. f'(x) = 2x sin(x) + x^2 cos(x)
  • Check: at x = π, f'(π) = 0 + π^2 × (-1) = -π^2 ≈ -9.87, which matches the slope of f measured numerically near π
  • Answer: f'(x) = 2x sin(x) + x^2 cos(x)
  • Common mistake: multiplying the two derivatives, 2x cos(x), which ignores the product rule

Worked example: a standard deviation

Written by hand in the Answer + steps layout, including the Note line for an ambiguous problem:

  • Problem: Find the standard deviation of 4, 8, 6, 5, 7
  • Note: the problem does not say sample or population; this solution assumes a sample
  • 1. Mean: (4 + 8 + 6 + 5 + 7)/5 = 30/5 = 6
  • 2. Squared deviations: 4, 4, 0, 1, 1; their sum is 10
  • 3. Sample variance: 10/(5 - 1) = 2.5
  • 4. Standard deviation: sqrt(2.5) ≈ 1.58
  • Check: the population version is sqrt(10/5) = sqrt(2) ≈ 1.41, slightly smaller, as it should be
  • Answer: s ≈ 1.58 for a sample; σ ≈ 1.41 if the data are the whole population

Sample problems by topic

Copy a problem, or press Try it to open it in the solver.

Arithmetic

  • Fractions

    Simplify 3/4 + 5/6 - 1/3

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  • Percentages

    What is 18% of 250, and what percent of 250 is 60?

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  • Order of operations

    Evaluate 8 + 2 × (15 - 3^2) / 4

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Algebra

  • Linear equation

    Solve for x: 3(x - 4) + 7 = 2x + 9

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  • Quadratic equation

    Solve x^2 - 6x + 4 = 0 and give exact and decimal answers.

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  • System of equations

    Solve the system: 2x + 3y = 12 and x - y = 1

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Geometry

  • Circle

    A circle has a circumference of 31.4 cm. Find its radius and its area.

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  • Right triangle

    A right triangle has legs of 9 cm and 12 cm. Find the hypotenuse, the area and the perimeter.

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  • Cylinder

    Find the volume and the surface area of a cylinder with radius 3 cm and height 10 cm.

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Trigonometry

  • Ladder

    A 6 m ladder leans against a wall at 70 degrees to the ground. How high up the wall does it reach?

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  • Trig equation

    Solve 2sin(x) - 1 = 0 for 0 ≤ x < 360 degrees.

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  • Identity

    Prove that (1 - cos^2(x))/sin(x) = sin(x).

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Calculus

  • Derivative

    Find the derivative of f(x) = x^2 sin(x).

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  • Definite integral

    Evaluate the integral from 0 to 2 of (3x^2 + 2x) dx.

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  • Limit

    Find the limit as x -> 0 of sin(3x)/x.

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Statistics

  • Standard deviation

    Find the mean and standard deviation of 4, 8, 6, 5, 7.

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  • Probability

    Two dice are rolled. What is the probability that the sum is at least 10?

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  • Counting

    How many ways can a committee of 3 be chosen from 10 people?

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Word problems

  • Two trains

    A train leaves a station at 9:00 a.m. traveling at 80 km/h. A second train leaves the same station at 10:00 a.m. on the same route at 100 km/h. At what time does the second train catch up, and how far from the station?

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  • Stacked discounts

    A jacket is reduced by 25% and then by another 10%. What single discount is that?

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  • Working together

    Ana can paint a room in 3 hours and Ben in 5 hours. How long do they take working together?

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Examples FAQ

Can I try these examples for free?

You can read every problem and both worked solutions for free. Solving a problem in the tool, including these samples, uses 1 credit from a purchased pack or plan; new accounts do not get free credits.

Are the worked solutions written by the AI?

No. The two worked solutions on this page were written by hand to show the layout the solver uses. Your own solutions are written by Claude Sonnet 5 and may be worded differently.

Which topic should I choose for a mixed problem?

Choose the topic of the main method. A word problem that leads to a quadratic fits Word problems; an area question that needs trigonometry fits Geometry or Trigonometry. The topic adds method rules but does not limit what the solver may use.

Solve a sample, then your own problem

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Last updated October 29, 2026

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