a. Separate the fractions 2/6, 2/5, 6/13, 1/25, 7/8, and 9/29 into two
categories: those that can be written as a terminating
decimal and those that cannot.
2/6 = 1/3 = 0.333333333333333333333333333333333333333333333333333..., no
2/5 = 0.4, yes, a terminating decimal
6/13 = 0.461538461538461538461538461538461538461538461538461538..., no
1/25 = 0.04, yes, a terminating decimal
7/8 = 0.975, yes, a terminating decimal
9/29 = 0.310344827586290689655172..., no
Write an explanation
of how you made your decisions.
First we reduce the proper fractions to their lowest terms.
That only involved reducing 2/6 to 1/3.
The only prime factors of denominators 3, 5, 25, and 8 are 2 and 5.
The denominators 6, 13, and 29 have other prime factors besides 2 and 5.
b. Form a conjecture about which fractions can be
expressed as terminating decimals.
If the only prime factors of the denominator of a non-zero common fraction,
when reduced to lowest terms are 2 and 5, then the fraction can be written
as a terminating decimal, otherwise it can only be written as a
repeating decimal.
c. Test your conjecture on the following fractions:6/12, 7/15, 28/140,
and 0/7
6/12 reduces to 1/2, and the denominator has no prime factor other
than 2 or 5. In fact it has only prime factor 2. So it can be
expressed as the terminating decimal .5
7/15 is already in lowest terms, but its denominator contains prime
factor 3, so its denominator does not have only prime factors 2 and 5.
So 7/15 cannot be expressed as a terminating decimal, but only the
repeating decimal .466666666666666666666666666...
28/140 reduces to 1/5, and the denominator has no prime factor other
than 2 or 5. In fact it has only prime factor 5. So it can be
expressed as the terminating decimal .2
0/7 That fraction equals 0, so it cannot be expressed as a
terminating decimal.
d. Use the ideas of equivalent fractions and common
multiples to verify your conjecture.
Any terminating decimal < 1 which has n digits can be written as
a fraction which has those n digits as a numerator and 10^n as
a denominator. 10^n has only 2 and 5 as prime factors, and if
the fraction is reducted to lowest terms, its denominator still
will not have any prime factors other than 2 and 5.
Edwin