One half, 0.5 and fifty percent are, mathematically, the exact same quantity, just written three different ways. A fraction expresses a value as one whole number divided by another, a decimal expresses it using place value after a decimal point, and a percentage expresses it as a proportion out of a hundred. None of these forms is more mathematically "true" than the others; each exists because it makes certain kinds of calculation or comparison genuinely easier than the other two, not because the underlying number actually changes depending on which form is used.
Each representation makes a genuinely different kind of task easier
A fraction makes exact division and precise ratio relationships transparent, one third is unambiguously one part out of three equal parts, in a way decimal approximation, 0.333 recurring, actually obscures. A decimal makes arithmetic operations, especially addition and comparison of size, considerably more straightforward, since decimals line up neatly by place value in a way fractions with different denominators don't. A percentage makes comparing proportions across genuinely different totals intuitive, since a percentage always expresses "out of a hundred" regardless of what the original total actually was, letting a reader compare a 15% pay rise against a 15% price increase without needing to know either underlying number first.
Converting between the three forms is exact, not approximate, whenever the fraction terminates cleanly
Converting between these three forms is a matter of exact, reversible arithmetic, dividing a fraction's numerator by its denominator gives the equivalent decimal, and multiplying a decimal by a hundred gives the equivalent percentage, at least whenever the underlying fraction actually terminates in decimal form. Some fractions, like one third, only terminate cleanly in fraction form, requiring an infinitely repeating decimal to represent exactly, which is exactly why choosing the right representation for a given task, keeping a fraction like one third in fraction form rather than rounding it to a decimal, can matter for genuine precision, not just convenience.
What we're still unsure about
That fractions, decimals and percentages are simply different, exactly interconvertible representations of the same underlying quantity is well established, foundational mathematics that's been taught consistently for centuries. What's more genuinely a matter of ongoing pedagogical debate is exactly which representation to introduce first and how much emphasis to place on fluent conversion between the three, since students who learn one representation well sometimes struggle to recognise the same value in a different form, and maths educators continue to use meaningfully different sequencing and teaching strategies to build that flexible, cross-representation fluency, rather than one universally agreed best approach existing.
This sits inside Fractions, Decimals & Percentages, one of seven topics in Arithmetic, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.