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7. DECIMALS, RATIO, PROPORTION, AND PERCENT
Every fraction can be written as a repeating decimal. Ts see why this is so,
5
consider . In dividing by 7, there are 7 possible remainders, 0 through 6. Thus
7
a remainder must repeat by the 7th division:
5
Example.
7
5
= 0.714285
7
Theorem (Fractions with Repeating, Nonterminating Decimal Represena
a
tations). Let
be a fraction in simplest form. Then
has a repeating
b
b
decimal representation that does not terminate if and only if b has a prime
factor other than 2 or 5.