# Math solver examples by topic

> 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.

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.

## At a glance

- **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

[Try a sample](https://mathsolvercalculator.app/create)

## What these examples show

- **Every topic**: Three problems each for arithmetic, algebra, geometry, trigonometry, calculus, statistics and word problems.
- **Common traps**: Fractions with unlike denominators, stacked discounts, degrees versus radians and sample versus population.
- **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. **Pick a problem**: Choose one from your topic, or one close to your homework.
2. **Try both levels**: Solve with Answer + steps, then read the Detailed teaching version of a hard one.
3. **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
- **Percentages:** What is 18% of 250, and what percent of 250 is 60?
- **Order of operations:** Evaluate 8 + 2 × (15 - 3^2) / 4

### Algebra

- **Linear equation:** Solve for x: 3(x - 4) + 7 = 2x + 9
- **Quadratic equation:** Solve x^2 - 6x + 4 = 0 and give exact and decimal answers.
- **System of equations:** Solve the system: 2x + 3y = 12 and x - y = 1

### Geometry

- **Circle:** A circle has a circumference of 31.4 cm. Find its radius and its area.
- **Right triangle:** A right triangle has legs of 9 cm and 12 cm. Find the hypotenuse, the area and the perimeter.
- **Cylinder:** Find the volume and the surface area of a cylinder with radius 3 cm and height 10 cm.

### Trigonometry

- **Ladder:** A 6 m ladder leans against a wall at 70 degrees to the ground. How high up the wall does it reach?
- **Trig equation:** Solve 2sin(x) - 1 = 0 for 0 ≤ x < 360 degrees.
- **Identity:** Prove that (1 - cos^2(x))/sin(x) = sin(x).

### Calculus

- **Derivative:** Find the derivative of f(x) = x^2 sin(x).
- **Definite integral:** Evaluate the integral from 0 to 2 of (3x^2 + 2x) dx.
- **Limit:** Find the limit as x -> 0 of sin(3x)/x.

### Statistics

- **Standard deviation:** Find the mean and standard deviation of 4, 8, 6, 5, 7.
- **Probability:** Two dice are rolled. What is the probability that the sum is at least 10?
- **Counting:** How many ways can a committee of 3 be chosen from 10 people?

### 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?
- **Stacked discounts:** A jacket is reduced by 25% and then by another 10%. What single discount is that?
- **Working together:** Ana can paint a room in 3 hours and Ben in 5 hours. How long do they take working together?


## Simple pricing, by the problem

- Starter: $7.99 / month, 100 credits
- Pro: $19.99 / month, 250 credits
- Tutor: $49.99 / month, 650 credits
- Mini: $4.99, 50 credits
- Plus: $14.99, 160 credits
- Max: $34.99, 420 credits

## 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

Open any example in the solver, compare both explanation levels and then try your own numbers.

_Last updated 2026-10-29_