Soal dan Pembahasan Operasi Bilangan Pecahan

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1. $\displaystyle \frac{2}{3}+5\frac{1}{3}-2\frac{5}{6}=....$
a. $\displaystyle 3\frac{1}{6}$ $\quad$ c. $\displaystyle -3\frac{1}{6}$
b. $\displaystyle 4\frac{1}{6}$ $\quad$ d. $\displaystyle -4\frac{1}{6}$
Pembahasan:

$\displaystyle =\frac{2}{3}_{\times 2}+\frac{1}{3}_{\times 2}-\frac{5}{6}+5-2$ (pisahkan bilangan tengah pecahan campuran)
$\displaystyle = 3+\frac{4}{6}+\frac{2}{6}-\frac{5}{6}$
$\displaystyle = 3+\frac{4+2-5}{6}=3+\frac{1}{6}=3\frac{1}{6}$

2. $\displaystyle 5-\frac{2}{5}\times 7+\frac{10}{11}=....$
a. $\displaystyle 1\frac{6}{55}$ $\quad$ c. $\displaystyle 3\frac{6}{55}$
b. $\displaystyle 2\frac{6}{55}$ $\quad$ d. $\displaystyle 4\frac{6}{55}$
Pembahasan:

$$=5-\frac{2\times7}{5}+\frac{10}{11}$$ $$=5-\frac{14}{5}+\frac{10}{11}$$ $$=5+\frac{-14\times11+10\times5}{5\times11}$$ $$=5+\frac{-154+50}{55}$$ $$=5+\frac{-104}{55}=5-\frac{104}{55}$$ $$=\frac{5\times55-104}{55}=\frac{275-104}{55}=\frac{171}{55}$$ $$=3\frac{6}{55}$$

3. $\displaystyle \frac{\frac{3}{5}\times \frac{25}{21}-2\frac{7}{3}}{\frac{4}{3}-10}=....$
a. $\displaystyle \frac{35}{91}$ $\quad$ c. $\displaystyle \frac{37}{91}$
b. $\displaystyle \frac{36}{91}$ $\quad$ d. $\displaystyle \frac{38}{91}$
Pembahasan:

$$=\frac{\frac{3\times 25}{5\times 21}+\frac{3\times(-2)-7}{3}}{\frac{4+3\times(-10)}{3}}$$ $$=\frac{\frac{3}{21}\times\frac{25}{5}-\frac{13}{3}}{-\frac{26}{3}}$$ $$=\frac{\frac{1}{7}\times 5-\frac{13}{3}}{-\frac{26}{3}}$$ $$=-\frac{\frac{5}{7}-\frac{13}{3}}{\frac{26}{3}}$$ $$=-\frac{\frac{5\times 3-13\times 7}{7\times 3}}{\frac{26}{3}}$$ $$=-\frac{-\frac{76}{21}}{\frac{26}{3}}$$ $$=\frac{76}{21}\times \frac{3}{26}$$ $$=\frac{76}{26}\times \frac{3}{21}$$ $$=\frac{38}{13} \times \frac{1}{7}=\frac{38}{91}$$

4. $\displaystyle -3\times \left(\frac{2}{5}+\frac{5}{2}-5\times \frac{1}{3}\right): -7\frac{4}{9}=....$
a. $\displaystyle \frac{333}{670}$ $\quad$ c. $\displaystyle \frac{335}{670}$
b. $\displaystyle \frac{334}{670}$ $\quad$ d. $\displaystyle \frac{336}{670}$
Pembahasan:

$$=-3\times \left(\frac{2\times 2+5\times 5}{5\times 2}-\frac{5}{3}\right): \frac{9\times (-7)-4}{9}$$ $$=-3\times \left(\frac{29}{10}-\frac{5}{3}\right):-\frac{67}{9}$$ $$=-3\times \frac{29\times 3-5\times 10}{10 \times 3}\times -\frac{9}{67}$$ $$=3\times \frac{87-50}{30}\times \frac{9}{67}$$ $$=3\times \frac{37}{30}\times \frac{9}{67}$$ $$=\frac{37}{10}\times \frac{9}{67}=\frac{37\times 9}{10 \times 67}$$ $$=\frac{333}{670}$$

5. $\displaystyle \left(-1\frac{2}{3}+5\frac{1}{6}\right)^{-2}\times \frac{7}{8}+\frac{1}{8}=....$
a. $\displaystyle \frac{13}{56}$ $\quad$ c. $\displaystyle -\frac{13}{56}$
b. $\displaystyle \frac{11}{56}$ $\quad$ d. $\displaystyle -\frac{11}{56}$
Pembahasan:

$$=\left(-1+5-\frac{4}{6}+\frac{1}{6}\right)^{-2}\times \frac{7}{8}+\frac{1}{8}$$ $$=\left(4+\frac{-4+1}{6}\right)^{-2}\times \frac{7}{8}+\frac{1}{8}$$ $$=\left(4-\frac{1}{2}\right)^{-2}\times \frac{7}{8}+\frac{1}{8}$$ $$=\left(\frac{7}{2}\right)^{-2} \times \frac{7}{8}+\frac{1}{8}$$ $$=\left(\frac{2}{7}\right)^{2} \times \frac{7}{8}+\frac{1}{8}$$ $$=\frac{2^2}{7^2} \times \frac{7}{8}+\frac{1}{8}$$ $$=\frac{4}{49} \times \frac{7}{8}+\frac{1}{8}$$ $$=\frac{4}{8}\times \frac{7}{49}+\frac{1}{8}$$ $$=\frac{1}{2}\times \frac{1}{7}+\frac{1}{8}$$ $$=\frac{1}{14}+\frac{1}{8}=\frac{1}{14}_{\times 4}+\frac{1}{8}_{\times 7}$$ $$=\frac{4+7}{56}=\frac{11}{56}$$

6. $\displaystyle \frac{\sqrt{\frac{2}{3}\times \frac{8}{3}}+ 1\frac{1}{2}}{\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\right)^{-1}}$
a. $\displaystyle 3\frac{61}{144}$ $\quad$ c. $\displaystyle 5\frac{61}{144}$
b. $\displaystyle 4\frac{61}{144}$ $\quad$ d. $\displaystyle 6\frac{61}{144}$
Pembahasan:

$$=\frac{\sqrt{\frac{16}{9}}\times \frac{8}{3}+\frac{3}{2}}{\left(\frac{1}{2}_{\times 4}+\frac{1}{4}_{\times 2}+\frac{1}{8}_{\times 1}\right)^{-1}}$$ $$=\frac{\frac{4}{3} \times \frac{8}{3}+\frac{3}{2}}{\left(\frac{4+2+1}{8}\right)^{-1}}$$ $$=\frac{\frac{32}{9}+\frac{3}{2}}{\frac{8}{7}}$$ $$=\frac{32\times 2+3\times 9}{9\times 2}\times \frac{7}{8}$$ $$=\frac{91}{18}\times \frac{7}{8}=\frac{91\times 7}{18 \times 8}$$ $$=\frac{637}{144}=4\frac{61}{144}$$

7. $\displaystyle \frac{\left(\frac{15}{8}-4\frac{2}{9}\right)^{-2}\times \frac{7}{5}}{-\frac{1}{3}+\frac{5}{2}}$
a. $\displaystyle \frac{217728}{1856465}$ $\quad$ c. $\displaystyle \frac{217726}{1856465}$
b. $\displaystyle \frac{217727}{1856465}$ $\quad$ d. $\displaystyle \frac{217725}{1856465}$
Pembahasan:

$$=\frac{\left(\frac{15}{8}+\frac{9\times (-4)-2}{9}\right)^{-2}\times \frac{7}{5}}{\frac{-1\times 2+3\times 5}{3\times 2}}$$ $$=\frac{\left(\frac{15}{8}-\frac{38}{9}\right)^{-2}\times \frac{7}{5}}{\frac{-2+15}{6}}$$ $$=\left(\frac{15\times 9 -38 \times 8}{8\times 9}\right)^{-2} \times \frac{7}{5}\times \frac{6}{13}$$ $$=\left(-\frac{169}{72}\right)^{-2} \times \frac{7 \times 6}{5\times 13}$$ ingat bahwa $(-)^2=(-).(-)=+$, sehingga $$=\frac{72^2}{169^2} \times \frac{42}{65}$$ $$=\frac{5184}{28561} \times \frac{42}{65}$$ $$=\frac{5184 \times 42}{28561 \times 65}$$ $$=\frac{217728}{1856465}$$

8. $\displaystyle \frac{1}{\frac{1}{10}+\frac{2}{100}+\frac{3}{1000}+\frac{4}{10000}}=.... $
a. $\displaystyle 7\frac{64}{617}$ $\quad$ c. $\displaystyle 9\frac{64}{617}$
b. $\displaystyle 8\frac{64}{617}$ $\quad$ d. $\displaystyle 10\frac{64}{617}$
Pembahasan:

$$=\frac{1}{\frac{1\times 1000+2\times 100+3\times 10+4\times 1}{10000}}$$ $$=\frac{10000}{1000+200+30+4}=\frac{10000}{1234}$$ $$=8\frac{64}{617}$$