Question

arati ca urmatoarele fractii sunt echivalente: 77/121, 21/33, 2121/3333 si 2163/3399​

Answer 1

$\frac{77}{121} = \frac{21}{33} = \frac{2121}{3333} = \frac{2163}{3399}$

Condiția este:

$77\cdot 33 - 21\cdot 121 = 2121\cdot 3399 - 2163\cdot 3333\\ \\ \Rightarrow 2541-2541 = 7209279 - 7209279\\ \\ \Rightarrow 0 = 0\,\,\,\,\checkmark$

Demonstrație:

$\frac{77}{121} - \frac{21}{33} = 0 =\frac{2121}{3333} - \frac{2163}{3399}\\ \\ \Rightarrow \frac{77}{121} - \frac{21}{33} = \frac{2121}{3333} - \frac{2163}{3399}\\ \\ \Rightarrow \frac{77\cdot 33-21\cdot 121}{121\cdot 33}= \frac{2121\cdot 3399-2163\cdot 3333}{3333\cdot 3399}\\ \\ \Rightarrow (77\cdot 33-21\cdot 121)\cdot 3333\cdot 3399 =\\ \\ ~\,\,\,\,\,\,\,=(2121\cdot 3399-2163\cdot 3333)\cdot 121\cdot 33\\ \\ \Rightarrow 0 \cdot 3333\cdot 3399= 0\cdot 121\cdot 33 \\ \\ \Rightarrow 0 = 0\,\,\,\,\text{(q.e.d.)}$