Question

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:i)13/3125ii)17/8iii)64/455iv)15/1600v)29/343vi)\frac{23}{2^{3}\times 5^{2}}vii)\frac{129}{2^{2}\times 5^{7}\times 7^{5}}viii)6/15ix)35/50x)77/210

Answer 1

Solution:

A number is terminating decimal only if the denominator has the prime factors as
${2}^{n} {5}^{m} \: \: n \: and \: m \: \epsilon Z^+$
$i) \frac{13}{3125} \\ \\ = > \frac{13}{ {5}^{5} }$
terminating decimal expansion.


$ii) \frac{17}{8} = \frac{17}{ {2}^{3} } \\ \\ terminating \: decimal \: expansion$

$iii) \frac{64}{455} \\ \\ = > \frac{ {2}^{6} }{5 \times 13 \times 7} \\ \\ non \: terminating \: repeating \: \\ decimal \: expansion$


$iv) \frac{15}{1600} \\ \\ = > \frac{3 \times 5}{ {2}^{4} \times 100} \\ \\ = > \frac{3 \times5 }{ {2}^{4} \times {(5 \times 2)}^{2} } \\ \\ = > \frac{3 }{ {2}^{6} \times {5} } \\ \\ terminating \: decimal \: expansion$

$v) \frac{29}{343} = \frac{29}{ {7}^{3} } \\ \\ non \: terminating \: repeating \: \\ decimal \: expansion$


$vi) \frac{23}{ {2}^{3} \times {5}^{2} } \\ \\ \: terminating \: decimal \: expansion$


$vii) \frac{129}{ {2}^{2} \times {5}^{7} \times {7}^{5} } \\ \\ non \: terminating \: repeating \: \\ decimal \: expansion$


$viii) \frac{6}{15} = \frac{3 \times 2}{3 \times 5} \\ \\ = \frac{2}{5} \\ \\\: terminating \: decimal \: expansion$


$ix) \frac{35}{50} = \frac{5 \times 7}{2 \times {5}^{2} } \\ \\ = \frac{7}{2 \times 5} \\ \\ \: terminating \: decimal \: expansion$


$x) \: \frac{77}{210} = \frac{11 \times 7}{2 \times 3 \times 5 \times 7} \\ \\ = \frac{11}{2 \times 3 \times 5} \\ \\ non \: terminating \: repeating \: \\ decimal \: expansion$

Hope it helps you.