Exercise 2: Why 0.5 Is Exact but 0.1 and 0.2 Are Not — Possible Solution
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WHAT DETERMINES WHETHER A BINARY EXPANSION TERMINATES
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A fraction's binary expansion terminates (has a finite number of
digits) exactly when the fraction, written in lowest terms, has a
denominator that is a pure power of 2 (1, 2, 4, 8, 16, ...). This is
the binary equivalent of the decimal rule that a fraction terminates
in decimal exactly when its denominator (in lowest terms) is only
made of the prime factors of 10, namely 2 and 5.

WHY 0.5 IS EXACT
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0.5 = 1/2. The denominator, 2, is exactly 2^1 - a pure power of two.
In binary this is simply 0.1 (one binary digit after the point,
representing 2^-1). There is no repeating pattern - the expansion
terminates after a single digit, so it fits perfectly into a
floating-point mantissa with room to spare and stores with zero
error.

WHY 0.1 AND 0.2 ARE NOT EXACT
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0.1 = 1/10, and 10 = 2 * 5. Because the denominator has a factor of
5 in it - a prime factor that is not 2 - the fraction cannot be
written as an exact power-of-two denominator no matter how it's
simplified. The same is true of 0.2 = 1/5, whose denominator is
purely 5, again not a power of 2. This chapter verified directly
that both 0.1 and 0.2 expand into the infinitely repeating binary
pattern 0011 rather than terminating. Since only a finite number of
mantissa bits (52 for a double) are available, that infinite pattern
must be cut off and rounded, which is exactly why 0.1 and 0.2 are
each stored as a close approximation rather than their true value.

THE GENERAL RULE
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A decimal fraction stores exactly in binary floating point if and
only if it can be written as (some integer) / 2^n for a non-negative
integer n - in other words, only sums of exact negative powers of 2
(1/2, 1/4, 1/8, 1/16, ...) are guaranteed exact. Any fraction whose
reduced denominator contains a prime factor other than 2 - which
includes the overwhelming majority of ordinary decimal fractions
like 0.1, 0.2, 0.3, and 0.7 - will repeat forever in binary and
therefore be stored only approximately.

WHY THIS WORKS AS AN ANSWER
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The explanation identifies the actual mathematical condition (prime
factors of the reduced denominator) rather than just restating that
"0.1 doesn't work and 0.5 does," applies that condition correctly to
both numbers, and generalizes it into a rule the reader could apply
to a new fraction without having to expand it by hand.