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Mathematics

Write the following in decimal form and say what kind of decimal expansion each has :

(i) 36100\dfrac{36}{100}

(ii) 111\dfrac{1}{11}

(iii) 4184\dfrac{1}{8}

(iv) 313\dfrac{3}{13}

(v) 211\dfrac{2}{11}

(vi) 329400\dfrac{329}{400}

Number System

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Answer

(i) 36100\dfrac{36}{100} = 0.36, terminating decimal expansion

(ii) 111\dfrac{1}{11}

11)0.090911)100‾11)22−99‾11x3−210011)x322−99‾11)x32222111)x3221111‾\begin{array}{l} \phantom{11)}{0.0909} \ 11\overline{\smash{\big)}\quad100\quad} \ \phantom{11)}\phantom{22}\underline{-99} \ \phantom{{11}x^3-2}100 \ \phantom{{11)}x^322}\underline{-99} \ \phantom{{11)}{x^32222}}1 \ \phantom{{11)}{x^322}}\overline{\phantom{1111}} \end{array}

The remainder 1 keeps repeating.
Hence, 111\dfrac{1}{11} = 0.09‾0.\overline{09}. This is a non-terminating repeating decimal expansion

(iii) 418=3384\dfrac{1}{8} = \dfrac{33}{8}

8)4.1258)33‾8)−32 ‾8)−108−−8‾8)−8208−−16‾8)−82408−8−40‾8−82208−811111‾\begin{array}{l} \phantom{8)}{\enspace 4.125} \ 8\overline{\smash{\big)}\enspace 33\quad} \ \phantom{8)}\underline{-32\space} \ \phantom{8)-}10 \ \phantom{8-}\underline{-8} \ \phantom{8)-8}20 \ \phantom{8-}\underline{-16} \ \phantom{8)-82}40 \ \phantom{8-8}\underline{-40\enspace} \ \phantom{8-822}0 \ \phantom{8-8}\overline{\phantom{11111}} \end{array}

Hence, 4184\dfrac{1}{8} = 4.125. This is a terminating decimal expansion.

(iv) 313\dfrac{3}{13}

8)0.23076923076913)30‾13)−26 ‾13)−408−−39‾8)−810013303−91‾13303333901330331−78‾13303333312013303333−117‾1330333333330133033333)−26‾1330333333340133033333311111‾\begin{array}{l} \phantom{8)}{\enspace 0.230769230769} \ 13\overline{\smash{\big)}\enspace 30\qquad\qquad\quad} \ \phantom{13)}\underline{-26\space} \ \phantom{13)-}40 \ \phantom{8-}\underline{-39} \ \phantom{8)-8}100 \ \phantom{13303}\underline{-91\enspace} \ \phantom{13303333}90 \ \phantom{1330331}\underline{-78\enspace} \ \phantom{133033333}120 \ \phantom{13303333}\underline{-117\enspace} \ \phantom{13303333333}30 \ \phantom{133033333)}\underline{-26\enspace} \ \phantom{13303333333}40 \ \phantom{1330333333}\overline{\phantom{11111}} \end{array}

Hence, 313\dfrac{3}{13} = 0.230769‾0.\overline{230769}. This is a non-terminating repeating decimal expansion.

(v) 211\dfrac{2}{11}

8)0.181811)20‾11)−11 ‾11)−908−−88‾8)−8)201330)−11‾1330333)90133033−88‾13303333213303311111‾\begin{array}{l} \phantom{8)}{\enspace 0.1818} \ 11\overline{\smash{\big)}\enspace 20\qquad} \ \phantom{11)}\underline{-11\space} \ \phantom{11)-}90 \ \phantom{8-}\underline{-88} \ \phantom{8)-8)}20 \ \phantom{1330)}\underline{-11\enspace} \ \phantom{1330333)}90 \ \phantom{133033}\underline{-88\enspace} \ \phantom{13303333}2 \ \phantom{133033}\overline{\phantom{11111}} \end{array}

Hence, 211=0.18‾\dfrac{2}{11} = 0.\overline{18}. This is a non-terminating repeating decimal expansion.

(vi) 329400\dfrac{329}{400} = 0.8225, terminating decimal expansion

400)0.8225400)3290‾400)−3200 ‾400)−900400)2−800‾400)2181000400)21−800‾400)212)2000400)21−2000‾400)211110400)211111111‾\begin{array}{l} \phantom{400)}{0.8225} \ 400\overline{\smash{\big)}\enspace 3290\quad} \ \phantom{400)}\underline{-3200\space} \ \phantom{400)-}900 \ \phantom{400)2}\underline{-800} \ \phantom{400)218}1000 \ \phantom{400)21}\underline{-800} \ \phantom{400)212)}2000 \ \phantom{400)21}\underline{-2000\enspace} \ \phantom{400)21111}0 \ \phantom{400)21}\overline{\phantom{1111111}} \end{array}

Hence, 329400\dfrac{329}{400} = 0.8225. This is a terminating decimal expansion.

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