Percentages & ratios
Percentage Change vs Percentage Difference
Learn why percentage change and percentage difference use different reference values, with worked examples and common mistakes.
7 min readReviewed 2026-08-13
Read guide Editorial category
Mathematics guides
Understand the methods behind arithmetic, percentages, averages, geometry, and other mathematical tools.
Practical guides
Averages & means
Weighted Average Explained
Learn when a weighted average is appropriate, how to calculate it, and why unequal weights change the result.
7 min readReviewed 2026-08-13
Read guide Triangle geometry
How to Solve a Triangle from Known Values
Choose between SSS, SAS, ASA, AAS, and SSA triangle-solving methods and recognize when information is insufficient or ambiguous.
8 min readReviewed 2026-08-13
Read guide Ratios & proportions
Ratios, Proportions, and Scaling in Real Situations
Understand what ratios preserve, when a proportion is justified, and how scaling changes quantities in recipes, drawings, mixtures, and comparisons.
7 min readReviewed 2026-08-13
Read guide Precision & measurement
Significant Figures, Precision, and Measurement Uncertainty
Learn what written significant figures communicate, why extra decimals do not create accuracy, and how rounding differs from uncertainty analysis.
8 min readReviewed 2026-08-13
Read guide Fractions & decimals
Using Fractions in Measurement, Sharing, and Scaling
Understand fractions as quantities, ratios, operators, and shares, and choose exact fraction arithmetic when decimals would hide the relationship.
7 min readReviewed 2026-08-13
Read guide Percentages
How to Calculate Percentages in Real Situations
Choose the correct whole, part, and percentage when comparing prices, scores, completion, populations, and measured quantities.
7 min readReviewed 2026-08-13
Read guide Percentages
Reverse Percentages: Finding the Original Value
Recover a starting value after a percentage increase or decrease and avoid the common mistake of reversing with simple subtraction.
6 min readReviewed 2026-08-13
Read guide Arithmetic
Order of Operations: Why Calculation Sequence Matters
Read grouped expressions consistently, understand exponent and sign conventions, and use parentheses to communicate the intended calculation.
7 min readReviewed 2026-08-13
Read guide Precision
How and When to Round Numbers
Choose a rounding place and rule that match the decision, preserve intermediate precision, and avoid hiding important boundary effects.
7 min readReviewed 2026-08-13
Read guide Number notation
Scientific Notation for Very Large and Small Numbers
Read, compare, calculate with, and report powers-of-ten notation without losing the scale or intended precision of a quantity.
6 min readReviewed 2026-08-13
Read guide Algebra
How to Solve a Linear Equation and Check the Result
Isolate an unknown while preserving equality, recognise exceptional cases, and translate a real situation into a defensible linear model.
7 min readReviewed 2026-08-13
Read guide Algebra
Understanding the Quadratic Formula and Discriminant
Interpret quadratic roots, use the discriminant to classify solutions, and connect algebraic answers with graphs and practical domains.
7 min readReviewed 2026-08-13
Read guide Geometry
The Pythagorean Theorem in Measurement and Layout
Use right-triangle geometry to find missing lengths, verify squareness, and understand when the theorem does not apply.
7 min readReviewed 2026-08-13
Read guide Plane geometry
Area vs Perimeter: Choosing the Correct Measurement
Distinguish boundary length from enclosed surface, select the right dimensions, and avoid assuming that one determines the other.
7 min readReviewed 2026-08-13
Read guide Solid geometry
Volume vs Surface Area in Practical Problems
Distinguish three-dimensional capacity from exposed boundary area and choose the geometry, units, and exclusions that match the task.
7 min readReviewed 2026-08-13
Read guide Averages & means
Which Mean Should You Use? Arithmetic, Geometric, Harmonic, or RMS
Choose a mean based on what is being combined rather than applying the familiar arithmetic average to every dataset.
7 min readReviewed 2026-08-13
Read guide Number fundamentals
Absolute Value as Distance, Magnitude, and Error
Understand absolute value as distance from zero and use it correctly for magnitude, difference, tolerance, and piecewise relationships.
6 min readReviewed 2026-08-13
Read guide Algebra
Solving Systems of Equations in Real Problems
Interpret intersections, choose substitution or elimination, and distinguish unique, inconsistent, and dependent linear systems.
7 min readReviewed 2026-08-13
Read guide Algebra
Understanding Inequalities and Their Solution Sets
Solve linear inequalities, reverse the relation when required, and communicate open, closed, bounded, and unbounded solution sets.
6 min readReviewed 2026-08-13
Read guide Algebra
Factoring Quadratic Expressions and Recognising Useful Patterns
Connect factors, roots, expansion, and the discriminant while distinguishing exact factorisation from decimal approximation.
7 min readReviewed 2026-08-13
Read guide Exponents
Exponent Rules and Why They Work
Understand powers as repeated multiplication and extend the rules carefully to zero, negative, and fractional exponents.
7 min readReviewed 2026-08-13
Read guide Logarithms
Understanding Logarithms as Inverse Exponents
Interpret a logarithm as an exponent, compare bases, and understand the domains and applications behind logarithmic scales.
7 min readReviewed 2026-08-13
Read guide Roots & radicals
Roots and Radicals: Exact Forms, Simplification, and Approximation
Interpret indexed roots, simplify perfect-power factors, and decide when an exact radical or decimal approximation is more useful.
7 min readReviewed 2026-08-13
Read guide Circle geometry
Circle Measurements: Radius, Diameter, Circumference, and Area
Connect the main measurements of a circle, keep linear and squared quantities distinct, and handle partial or measured circles honestly.
7 min readReviewed 2026-08-13
Read guide Trigonometry
Sine, Cosine, and Tangent in Practical Geometry
Interpret the primary trigonometric ratios, choose degrees or radians, and connect angle calculations with right-triangle and periodic models.
7 min readReviewed 2026-08-13
Read guide Coordinate geometry
Slope and the Equation of a Line
Interpret rise over run, distinguish vertical and horizontal lines, and connect points with slope-intercept and point-slope equations.
7 min readReviewed 2026-08-13
Read guide Coordinate geometry
Distance and Midpoint Between Two Coordinates
Measure straight-line separation and the halfway point between coordinates while keeping dimensions, units, and coordinate assumptions explicit.
7 min readReviewed 2026-08-13
Read guide Angles
Degrees, Radians, Gradians, and Turns Explained
Convert angular units, understand why radians connect naturally with arc length, and avoid calculator-mode errors.
6 min readReviewed 2026-08-13
Read guide Counting
Permutations vs Combinations: Does Order Matter?
Choose the correct counting model by deciding whether order and repetition create genuinely different outcomes.
7 min readReviewed 2026-08-13
Read guide Number theory
Prime Factorisation and the Structure of Whole Numbers
Break positive integers into prime powers and use that structure to understand divisors, common factors, and exact arithmetic.
7 min readReviewed 2026-08-13
Read guide Number theory
GCF vs LCM: Choosing the Right Common-Number Tool
Distinguish greatest common factors from least common multiples and recognise which one fits grouping or synchronisation problems.
6 min readReviewed 2026-08-13
Read guide Number theory
Modular Arithmetic: Remainders, Clocks, and Repeating Cycles
Understand congruence as sameness of remainder and use modular arithmetic to reason about repeating integer cycles.
7 min readReviewed 2026-08-13
Read guide Sequences & series
Arithmetic vs Geometric Sequences: Difference or Ratio?
Identify linear and multiplicative sequence patterns, interpret their parameters, and avoid extrapolating beyond the model.
7 min readReviewed 2026-08-13
Read guide Binary numbers
How Binary Numbers Represent Values
Read positional base-two notation, distinguish representations from values, and understand unsigned and signed bit patterns.
7 min readReviewed 2026-08-13
Read guide Linear algebra
Matrix Operations and What Their Results Mean
Understand matrix dimensions, multiplication order, determinants, and inverses without treating arrays as ordinary numbers.
7 min readReviewed 2026-08-13
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