Free lesson · Math & notation

Start with numbers, variables and an equals sign

Open interactive lesson Practice calculations Explore labs

Start with the idea

A variable names a number.
An equation says two expressions have the same value.

Symbols, units & horizon

x: unknown dimensionless number
2: constant multiplier
3: constant added amount
=: equal value
⇒: the following result is implied
numerator: entire expression above the fraction bar
denominator: entire nonzero expression below the fraction bar

When and why to use this

Use rearrangement before solving for an unknown price, quantity or model parameter. Track units on each side.

Read a fraction before solving with it

A fraction is a division written vertically. The numerator is above the bar; the denominator is below it.
For positive whole-number denominators, divide one whole into that many equal parts. The numerator counts how many parts are selected.
Three selected parts out of four give three quarters. Six selected eighths cover the same amount.
More selected parts than fit in one whole produce an improper fraction. Five quarters are one whole and one extra quarter.
The denominator cannot be zero. A fraction bar also groups the complete numerator and denominator.

\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} = 0.75

Fraction identity · equal parts

Equivalent fractions: split each piece in two

Begin with three selected quarters. Split every quarter into two equal pieces.
The whole now has eight pieces; the selected amount now has six. Neither the whole nor the selected amount changed.
Algebraically, multiply by one: $\frac{3}{4} \times \frac{2}{2} = \frac{6}{8}$ .
Both ratios give .75 as a decimal. Changing only the numerator or only the denominator would change the value.

Work it by hand

For five quarters, separate a whole: $\frac{5}{4} = \frac{4}{4} + \frac{1}{4} = 1.25$ .

A number is a value: 3.
A variable is a label for a value: x.
Multiplication repeats a quantity: 2x means 2 × x.
An equals sign asserts equal values on both sides.
Solve by undoing operations in reverse order.
Keep an equality true: apply the same operation to both sides.

2 x + 3 = 11 ⟹ x = 4

Core rule · definition and worked arithmetic

Start with numbers, variables and an equals sign

Subtract 3 from both sides: $2 x + 3 - 3 = 11 - 3$ .
Simplify: $2 x = 8$ .
Divide both sides by 2: $x = 4$ .
Check by substitution: $2 (4) + 3 = 11$ .

Work it by hand

The original left side at x=4 is 8+3=11. The equality holds.

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Textbook reference · checked 12 September 2026: OpenStax: visualize fractions. This is foundational arithmetic, not a market-performance result.

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def solve_linear(a, b, target):
    if a == 0: raise ValueError("Nonzero multiplier required")
    return (target-b)/a

print(solve_linear(2,3,11))

def fraction_value(numerator, denominator):
    if denominator == 0:
        raise ValueError("A fraction needs a nonzero denominator")
    return numerator / denominator

print(fraction_value(3,4), fraction_value(6,8))

Continue learning

Math & Notation Essentials — all lessons

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations