Question

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion: 3/8

Answer 1

AnswEr :

☯ Theorem Used : Let x be a rational number whose decimal expansion terminates, then x can be expressed in the form of p/q where p and q are co - prime and prime factorization of q is in the form of $\sf\ 2^n5^m$ where m and n are integers.

$\rule{120}2$

$\underline{\bigstar\:\textsf{Let's \: head \: to \: question \: now:}} \\\\\\\normalsize\ : \implies\sf\ Fraction = \frac{3}{8} \\\\\\\normalsize\ : \implies\sf\frac{3}{8} = \frac{3}{2^3}\\\\\\\normalsize\ : \implies\sf\frac{3}{8} = \frac{3 \times\ 5^3 }{2^3 \times\ 5^3}\\\\\\\normalsize\ : \implies\sf\frac{3}{8} =\frac{3 \times\ 125}{(2 \times\ 5)^3}\\\\\\\normalsize\ : \implies\sf\frac{3}{8} = \frac{375}{10^3}\\\\\\\normalsize\ : \implies\sf\frac{3}{8} = \frac{375}{10000}\\\\\\\normalsize\ : \implies\sf\frac{3}{8} = 0.375\\\\\\\normalsize\ : \implies{\underline{\boxed{\sf \red{\frac{3}{8} = 0.375 = Terminating}}}}$