Question

Cho \(\frac{a+5}{a-5}=\frac{b+6}{b-6}\)\(\) (a khác 5, b khác 6). Chứng minh rằng \(\frac{a}{b}=\frac{5}{6}\)

Answer 1

\(\frac{a+5}{a-5}=\frac{b+6}{b-6}\Rightarrow\left(a+5\right)\left(b-6\right)=\left(a-5\right)\left(b+6\right)\)

\(\Rightarrow ab-6a+5b-30=ab+6a-5b-30\)

\(\Rightarrow5b=6a\Leftrightarrow\frac{a}{b}=\frac{5}{6}\)

 (Đpcm)

Answer 2

Từ \(\frac{a+5}{a-5}=\frac{b+6}{b-6}\Rightarrow\frac{b-6}{a-5}=\frac{b+6}{a+5}\)

Áp dụng t/c dãy tỉ số bằng nhau : 

\(\frac{b-6}{a-5}=\frac{b+6}{a+5}=\frac{\left(b+6\right)-\left(b-6\right)}{\left(a+5\right)-\left(a-5\right)}=\frac{12}{10}=\frac{6}{5}\)

\(\Rightarrow5\left(b-6\right)=6\left(a-5\right)\Leftrightarrow5b=6a\Leftrightarrow\frac{a}{b}=\frac{5}{6}\)

Answer 3

Answer 4

Từ \(\frac{a+5}{a-5}\) = \(\frac{b+6}{b-6}\) => \(\frac{b-6}{a-5}\) = \(\frac{b+6}{a+5}\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có:

\(\frac{b-6}{a-5}=\) \(\frac{b+6}{a+5}\) = \(\frac{\left(b+6\right)-\left(b-6\right)}{\left(a+5\right)-\left(a-5\right)}\) = \(\frac{12}{10}\) = \(\frac{6}{5}\) 

=> 5 ( b-6 ) = 6 ( a - 5 ) 

<=> 5b - 6a <=> \(\frac{a}{b}\) = \(\frac{5}{6}\) 

Vậy \(\frac{a}{b}\)= \(\frac{5}{6}\)