Question

sa se verifice egalitățile:

d
e
f
va roooog​

Answer 1

$\displaystyle d)~\frac{^{4)}5}{9} +\frac{^{3)}1}{12} +\frac{^{2)}7}{18} =^{36)}1+\frac{1}{36} \Rightarrow \frac{4 \cdot 5+3 \cdot 1+2 \cdot 7}{36} =\frac{36 \cdot 1+1}{36} \Rightarrow \\\\ \Rightarrow \frac{20+3+14}{36} =\frac{36+1}{36}\Rightarrow \frac{37}{36} =\frac{37}{36} \rightarrow Adevarat.$

$\displaystyle e)~^{30)}4+\frac{^{10)}2}{3} +\frac{^{2)}5}{15} +\frac{^{15)}1}{2} =\frac{3}{6} +2+\Bigg(\frac{^{2)}13}{12} +\frac{10}{24} \Bigg) \Rightarrow \\\\ \Rightarrow \frac{30 \cdot 4+10 \cdot 2+2 \cdot 5+15 \cdot 1}{30} =\frac{3}{6} +2+\frac{2 \cdot 13+10}{24} \Rightarrow \\\\ \Rightarrow \frac{120+20+10+15}{30} =\frac{3}{6} +2+\frac{26+10}{24} \Rightarrow \frac{165}{30} =\frac{^{4)}3}{6} +^{24)}2+\frac{36}{24} \Rightarrow$

$\displaystyle \Rightarrow\frac{165}{30}=\frac{4\cdot3+24\cdot2+36}{24} \Rightarrow \frac{165}{30}=\frac{12+48+36}{24}\Rightarrow \frac{^{4)}165}{30}= \frac{^{5)}96}{24} \Rightarrow \\\\ \Rightarrow \frac{4 \cdot 165}{30} =\frac{5 \cdot 96}{24} \Rightarrow \frac{660}{120} =\frac{480}{120} \rightarrow Fals.$

$\displaystyle f)~\Bigg(\frac{^{7)}1}{10} +\frac{^{2)}1}{35} \Bigg)+\Bigg(~^{14)}2+\frac{1}{14} \Bigg)=\Bigg(~^{30)}2+\frac{3}{30} \Bigg)+\Bigg(\frac{^{4)}1}{35} +\frac{^{5)}2}{28} \Bigg) \Rightarrow \\\\ \Rightarrow \frac{7 \cdot 1+2 \cdot 1}{70} +\frac{14 \cdot 2+1}{14}=\frac{30\cdot2+3}{30}+\frac{4\cdot 1+5 \cdot 2}{140} \Rightarrow \\\\ \Rightarrow \frac{7+2}{70}+\frac{28+1}{14}=\frac{60+3}{30} +\frac{4+10}{140} \Rightarrow \frac{9}{70}+\frac{29}{14}=\frac{63^{(3}}{30} +\frac{14^{(14}}{140}\Rightarrow$

$\displaystyle \Rightarrow \frac{9}{70} +\frac{^{5)}29}{14} =\frac{21}{10} +\frac{1}{10} \Rightarrow \frac{9+5 \cdot 29}{70} =\frac{21+1}{10} \Rightarrow \frac{9+145}{70} =\frac{^{7)}22}{10} \Rightarrow \\\\ \Rightarrow \frac{154}{70} =\frac{7 \cdot 22}{10} \Rightarrow \frac{154}{70} =\frac{154}{70} \rightarrow Adevarat.$