Without actual division, find which of the following rational numbers have terminating decimal expansion. (i) 7/128 (ii) 21/15 (iii) 219/2200

Question

Without actual division, find which of the following rational numbers have terminating decimal expansion. 

(i) \(\frac{7}{128}\)

(ii) \(\frac{21}{15}\)

(iii) \(\frac{219}{2200}\)

Answer 1

(i) \(\frac{7}{128}\)

So \(\frac{7}{128}\) = \(\frac{7}{2^75^0}\)

This of the form 4m, n ∈ W 

So \(\frac{7}{128}\) has a terminating decimal expansion

(ii) \(\frac{21}{15}\)

So \(\frac{21}{15}\) has a terminating decimal expansion.

(iii) \(\frac{219}{2200}\)

\(\frac{219}{2200}\) = \(\frac{219}{2^35^211^1}\)

∴ This is not of the form \(\frac{p}{2^m5^n}\)

So \(\frac{219}{2200}\) has a non-terminating recurring decimal expansion.