Dupa ce am am vorbit despre adunarea numerelor rationale, astazi vom vorbi despre scaderea numerelor rationale.
Dupa cum stim de la adunarea numerelor rationale ca adunarea a doua numere rationale este tot un numar rational asa si diferenta a doua numere rationale este tot un numar rational. Incepem prin a rezolva cateva exercitii:
1) Efectuati:
a) ![<br /> -1\frac{3}{7}-\left[\left(-\frac{5}{42}\right)-\left(-\frac{3}{14}\right)-\left(+\frac{4}{21}\right)\right]=\\<br /> -\frac{1\cdot 7+3}{7}-\left(-\frac{5}{42}+\frac{3}{14}-\frac{4}{21}\right)=\\<br /> -\frac{10}{7}-\left(\frac{-1\cdot 5+3\cdot 3-2\cdot 4}{42}\right)=\frac{10}{7}-\left(\frac{-5+9-8}{42}\right)=<br /> -\frac{10}{7}-\left(\frac{-4}{42}\right)=-\frac{10}{7}-\left(\frac{-2}{21}\right)=-\frac{10}{7}+\frac{2}{21}=\frac{-10\cdot 3+1\cdot 2}{21}=\frac{-30+2}{21}=\frac{-28}{21}=-\frac{4}{3}<br />](https://s0.wp.com/latex.php?latex=%3Cbr+%2F%3E+-1%5Cfrac%7B3%7D%7B7%7D-%5Cleft%5B%5Cleft%28-%5Cfrac%7B5%7D%7B42%7D%5Cright%29-%5Cleft%28-%5Cfrac%7B3%7D%7B14%7D%5Cright%29-%5Cleft%28%2B%5Cfrac%7B4%7D%7B21%7D%5Cright%29%5Cright%5D%3D%5C%5C%3Cbr+%2F%3E+-%5Cfrac%7B1%5Ccdot+7%2B3%7D%7B7%7D-%5Cleft%28-%5Cfrac%7B5%7D%7B42%7D%2B%5Cfrac%7B3%7D%7B14%7D-%5Cfrac%7B4%7D%7B21%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+-%5Cfrac%7B10%7D%7B7%7D-%5Cleft%28%5Cfrac%7B-1%5Ccdot+5%2B3%5Ccdot+3-2%5Ccdot+4%7D%7B42%7D%5Cright%29%3D%5Cfrac%7B10%7D%7B7%7D-%5Cleft%28%5Cfrac%7B-5%2B9-8%7D%7B42%7D%5Cright%29%3D%3Cbr+%2F%3E+-%5Cfrac%7B10%7D%7B7%7D-%5Cleft%28%5Cfrac%7B-4%7D%7B42%7D%5Cright%29%3D-%5Cfrac%7B10%7D%7B7%7D-%5Cleft%28%5Cfrac%7B-2%7D%7B21%7D%5Cright%29%3D-%5Cfrac%7B10%7D%7B7%7D%2B%5Cfrac%7B2%7D%7B21%7D%3D%5Cfrac%7B-10%5Ccdot+3%2B1%5Ccdot+2%7D%7B21%7D%3D%5Cfrac%7B-30%2B2%7D%7B21%7D%3D%5Cfrac%7B-28%7D%7B21%7D%3D-%5Cfrac%7B4%7D%7B3%7D%3Cbr+%2F%3E+&bg=ffffff&fg=000&s=0&c=20201002)
Pentru a calcula exercitiul de mai sus am introdus intregii in fractii unde a fost nevoie, iar apoi am efectuat calculele din paranteza cu mare grija sa nu gresim semnele, am adus la acelasi numitor prima data in paranteza iar apoi primul termen cu ce am obtinut din paranteza, rezultatul obtinut l-am simplificat prin 7.
b)
![<br /> \left[1,3(5)-0,0(2)+0,(6)\right]-\left(1\frac{7}{15}-\frac{1}{5}\right)=\\<br /> \left(\frac{135-13}{90}-\frac{2}{90}+\frac{6}{9}\right)-\left(\frac{22}{15}-\frac{1}{5}\right)=\\<br /> \left(\frac{122}{90}-\frac{1}{45}+\frac{2}{3}\right)-\left(\frac{1\cdot 22-3\cdot 1}{15}\right)=\\<br /> \left(\frac{122\cdot 1-2\cdot 1+30\cdot 2}{90}\right)-\left(\frac{22-3}{15}\right)=\\<br /> \left(\frac{122-2+60}{90}\right)-\frac{19}{15}=\frac{180}{90}-\frac{19}{15}=\\<br /> 2-\frac{19}{15}=\frac{15\cdot 2-1\cdot19}{15}=\frac{30-19}{15}=\frac{11}{15}<br />](https://s0.wp.com/latex.php?latex=%3Cbr+%2F%3E+%5Cleft%5B1%2C3%285%29-0%2C0%282%29%2B0%2C%286%29%5Cright%5D-%5Cleft%281%5Cfrac%7B7%7D%7B15%7D-%5Cfrac%7B1%7D%7B5%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B135-13%7D%7B90%7D-%5Cfrac%7B2%7D%7B90%7D%2B%5Cfrac%7B6%7D%7B9%7D%5Cright%29-%5Cleft%28%5Cfrac%7B22%7D%7B15%7D-%5Cfrac%7B1%7D%7B5%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B122%7D%7B90%7D-%5Cfrac%7B1%7D%7B45%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Cright%29-%5Cleft%28%5Cfrac%7B1%5Ccdot+22-3%5Ccdot+1%7D%7B15%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B122%5Ccdot+1-2%5Ccdot+1%2B30%5Ccdot+2%7D%7B90%7D%5Cright%29-%5Cleft%28%5Cfrac%7B22-3%7D%7B15%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B122-2%2B60%7D%7B90%7D%5Cright%29-%5Cfrac%7B19%7D%7B15%7D%3D%5Cfrac%7B180%7D%7B90%7D-%5Cfrac%7B19%7D%7B15%7D%3D%5C%5C%3Cbr+%2F%3E+2-%5Cfrac%7B19%7D%7B15%7D%3D%5Cfrac%7B15%5Ccdot+2-1%5Ccdot19%7D%7B15%7D%3D%5Cfrac%7B30-19%7D%7B15%7D%3D%5Cfrac%7B11%7D%7B15%7D%3Cbr+%2F%3E+&bg=ffffff&fg=000&s=0&c=20201002)
La exercitiul b) am transformat fractiile periodice simple si mixte in fractii ordinare, am simplificat pe unde s-a putut, apoi am adus la acelasi numitor (am gasit numitorul comun) in fiecare parnteza rezultatele obtinute din cele doua paranteze le-am gasit numitorul comun si am efectuat calculele obtinand o fractie subunitara (numaratorul mai mic decat numitorul).
c)
![\left[2,08(3)-3\frac{5}{6}\right]-\left(3\frac{3}{4}-2\frac{1}{8}-2\frac{1}{6}\right)=\\<br /> \left(\frac{2083-208}{900}-\frac{3\cdot 6+5}{6}\right)-\left(\frac{3\cdot 4+3}{4}-\frac{2\cdot 8+1}{8}-\frac{2\cdot 6+1}{6}\right)=\\<br /> \left(\frac{1875}{900}-\frac{23}{6}\right)-\left(\frac{15}{4}-\frac{17}{8}-\frac{13}{6}\right)=\\<br /> \left(\frac{75}{36}-\frac{23}{6}\right)-\left(\frac{6\cdot 15-3\cdot 17-4\cdot 13}{24}\right)=\\<br /> \left(\frac{25}{12}-\frac{23}{6}\right)-\left(\frac{90-51-52}{24}\right)=\\<br /> \frac{1\cdot 25-2\cdot 23}{12}-\left(\frac{-13}{24}\right)=\frac{25-46}{12}-\left(-\frac{13}{24}\right)=\\<br /> \frac{-21}{12}+\frac{13}{24}=\frac{2\cdot(-21)+1\cdot 13}{24}=\frac{-42+13}{24}=\frac{-29}{24}=-\frac{29}{24}=-1\frac{5}{24}](https://s0.wp.com/latex.php?latex=+%5Cleft%5B2%2C08%283%29-3%5Cfrac%7B5%7D%7B6%7D%5Cright%5D-%5Cleft%283%5Cfrac%7B3%7D%7B4%7D-2%5Cfrac%7B1%7D%7B8%7D-2%5Cfrac%7B1%7D%7B6%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B2083-208%7D%7B900%7D-%5Cfrac%7B3%5Ccdot+6%2B5%7D%7B6%7D%5Cright%29-%5Cleft%28%5Cfrac%7B3%5Ccdot+4%2B3%7D%7B4%7D-%5Cfrac%7B2%5Ccdot+8%2B1%7D%7B8%7D-%5Cfrac%7B2%5Ccdot+6%2B1%7D%7B6%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B1875%7D%7B900%7D-%5Cfrac%7B23%7D%7B6%7D%5Cright%29-%5Cleft%28%5Cfrac%7B15%7D%7B4%7D-%5Cfrac%7B17%7D%7B8%7D-%5Cfrac%7B13%7D%7B6%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B75%7D%7B36%7D-%5Cfrac%7B23%7D%7B6%7D%5Cright%29-%5Cleft%28%5Cfrac%7B6%5Ccdot+15-3%5Ccdot+17-4%5Ccdot+13%7D%7B24%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cleft%28%5Cfrac%7B25%7D%7B12%7D-%5Cfrac%7B23%7D%7B6%7D%5Cright%29-%5Cleft%28%5Cfrac%7B90-51-52%7D%7B24%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cfrac%7B1%5Ccdot+25-2%5Ccdot+23%7D%7B12%7D-%5Cleft%28%5Cfrac%7B-13%7D%7B24%7D%5Cright%29%3D%5Cfrac%7B25-46%7D%7B12%7D-%5Cleft%28-%5Cfrac%7B13%7D%7B24%7D%5Cright%29%3D%5C%5C%3Cbr+%2F%3E+%5Cfrac%7B-21%7D%7B12%7D%2B%5Cfrac%7B13%7D%7B24%7D%3D%5Cfrac%7B2%5Ccdot%28-21%29%2B1%5Ccdot+13%7D%7B24%7D%3D%5Cfrac%7B-42%2B13%7D%7B24%7D%3D%5Cfrac%7B-29%7D%7B24%7D%3D-%5Cfrac%7B29%7D%7B24%7D%3D-1%5Cfrac%7B5%7D%7B24%7D&bg=ffffff&fg=000&s=0&c=20201002)
La exercitiul c) am transformat fractiile periodice mixte in fractii ordinare, am simplificat pe unde am putut pentru a ne simplifica calculele, iar apoi am adus la acelasi numitor si am calculat folosind regulile de calcul cu numere intregi, iar apoi am scos intregii din fractie.
Deci foarte important sa calculam corect, sa stim regulile de calcul cu numere intregi.
