Question

If $x=\frac{4}{7},y=\frac{1}{2}$ and $z=(-\frac{3}{4})$, then verify the following:
(i) $x÷y\ne y÷x$
(ii) $(x÷y)÷z\ne x÷(y÷z)$
(iii) $x÷(y+z)\ne x÷y+x÷z$
(iv) $x÷(y-z)\ne x÷y-x÷z$

Answer 1

$\left(i\right)x÷y=\frac{4}{7}÷\frac{1}{2}=\frac{4}{7}\times \frac{2}{1}=\frac{8}{7}$

$y÷x=\frac{1}{2}÷\frac{4}{7}=\frac{1}{2}\times \frac{7}{4}=\frac{7}{8}$

Hence $x÷y\ne y÷x$

$\left(ii\right)(x÷y)÷z=\frac{8}{7}÷(-\frac{3}{4})=\frac{8}{7}\times (-\frac{4}{3})=-\frac{32}{21}$

$x÷(y÷z)=\frac{4}{7}÷(\frac{1}{2}÷(-\frac{3}{4}))=\frac{4}{7}÷(\frac{1}{2}\times (-\frac{4}{3}))=\frac{4}{7}÷(-\frac{2}{3})=\frac{4}{7}\times (-\frac{3}{2})=-\frac{6}{7}$

Hence $(x÷y)÷z\ne x÷(y÷z)$

$\left(iii\right)x÷(y+z)=\frac{4}{7}÷(\frac{1}{2}+(-\frac{3}{4}))=\frac{4}{7}÷(\frac{1}{2}-\frac{3}{4})=\frac{4}{7}÷\left(\frac{2-3}{4}\right)=\frac{4}{7}÷(-\frac{1}{4})=\frac{4}{7}\times (-\frac{4}{1})=-\frac{16}{7}$

$x÷y+x÷z=\frac{8}{7}+\frac{4}{7}÷(-\frac{3}{4})=\frac{8}{7}+\frac{4}{7}\times (-\frac{4}{3})=\frac{8}{7}-\frac{16}{21}=\frac{24-16}{21}=\frac{8}{21}$

Hence $x÷(y+z)\ne x÷y+x÷z$

$\left(iv\right)x÷(y-z)=\frac{4}{7}÷(\frac{1}{2}-(-\frac{3}{4}))=\frac{4}{7}÷(\frac{1}{2}+\frac{3}{4})=\frac{4}{7}÷\left(\frac{2+3}{4}\right)=\frac{4}{7}÷\frac{5}{4}=\frac{4}{7}\times \frac{4}{5}=\frac{16}{35}$

$x÷y-x÷z=\frac{8}{7}-(-\frac{16}{21})=\frac{8}{7}+\frac{16}{21}=\frac{24+16}{21}=\frac{40}{21}$

Hence $x÷(y-z)\ne x÷y-x÷z$