Maths SolverArithmetic is the part of mathematics most people think they finished learning around age twelve, and the part that, in my experience, causes the most quiet mistakes in everyone’s later work. A surprising number of wrong answers in algebra, geometry, and calculus are not algebra, geometry, or calculus mistakes — they are arithmetic mistakes that the rest of the work cannot recover from. This section covers the three operations that most often go subtly wrong: fractions, percentages, and the greatest common divisor and least common multiple.
Each piece is short by the standards of the rest of the site, because the topics are short. The aim is not to inflate them but to make sure the methods are clearly stated and that the calculator on each page shows enough of the working that you can use it as a check.
Articles in this section
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Simplifying fractions, properly
The greatest common divisor, why dividing by it gives the lowest terms in one step, and how to handle improper fractions and mixed numbers without losing track of the sign.
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Percentages: the three things you can be asked
“What is X% of Y?”, “X is what percent of Y?” and “X is P% of what?” Each one is one line of algebra, but mixing them up is the most common mistake on percentage questions. With a calculator that does all three.
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GCD and LCM, with the Euclidean algorithm
The 2,300-year-old algorithm for the greatest common divisor that is still the fastest one we know, why $\mathrm{gcd}(a,b) \times \mathrm{lcm}(a,b) = a \times b$, and how both quantities turn up whenever you have to add fractions or simplify ratios.
Where these come up later
Adding two fractions cleanly requires the LCM of the denominators. Simplifying a ratio — in chemistry, in cooking, in odds — requires the GCD. Reading a salary increase, an inflation figure, or any piece of personal-finance maths requires the percentage methods. None of these are flashy topics, but you use them constantly, and the few minutes spent making sure you have the methods right pays back for years.