MUSIC THEORY FUNDAMENTALS - Chapter 2, Exercise 1 Solution
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Proportional Halving from Whole Note to Eighth Note

PROBLEM
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If a whole note lasts 4 beats, how many beats does a single eighth note
last? Show the proportional halving from whole note down to eighth note
step by step.

SOLUTION
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Each note value is exactly half the duration of the one before it:

    Whole note   = 4 beats
    Half note    = 4 / 2 = 2 beats
    Quarter note = 2 / 2 = 1 beat
    Eighth note  = 1 / 2 = 0.5 beats

ANSWER: A single eighth note lasts 0.5 beats (half a beat).

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WHY THIS WORKS AS AN ANSWER

The chapter established that note values follow a strict binary halving
system: whole (1), half (1/2), quarter (1/4), eighth (1/8), sixteenth
(1/16), each fraction of the whole note's own duration. Since the whole
note here is defined as 4 beats, an eighth note - being 1/8 of a whole
note - is 4 x (1/8) = 0.5 beats. Working through it one halving step at a
time (as shown above) arrives at the identical answer, and demonstrates
why the system is called "binary": each step is a division by exactly 2,
never any other number.