In which of the following tables x and y vary directly?

Solution:

If $x$ and $y$ vary directly, the ratio of the corresponding values of $x$ and $y$ remain $s$ constant.

(i)

$\frac{x}{y}=\frac{7}{21}=\frac{1}{3}$

$\frac{x}{y}=\frac{9}{27}=\frac{1}{3}$

$\frac{x}{y}=\frac{13}{39}=\frac{1}{3}$

$\frac{x}{y}=\frac{21}{63}=\frac{1}{3}$

$\frac{x}{y}=\frac{25}{75}=\frac{1}{3}$

In all the cases, the ratio is the same. Therefore, $x$ and $y$ vary directly.

(ii)

$\frac{x}{y}=\frac{10}{5}=2$

$\frac{x}{y}=\frac{20}{10}=2$

$\frac{x}{y}=\frac{30}{15}=2$

$\frac{x}{y}=\frac{40}{20}=2$

$\frac{x}{y}=\frac{46}{23}=2$

(iii)

$\frac{x}{y}=\frac{2}{6}=\frac{1}{3}$

$\frac{x}{y}=\frac{3}{9}=\frac{1}{3}$

$\frac{x}{y}=\frac{4}{12}=\frac{1}{3}$

$\frac{x}{y}=\frac{5}{17}=\frac{5}{17}$

$\frac{x}{y}=\frac{6}{20}=\frac{3}{10}$

In all the cases, the ratio is not the same. Therefore, $x$ and $y$ do not vary directly.

(iv)

$\frac{x}{y}=\frac{1^{2}}{1^{3}}=1$

$\frac{x}{y}=\frac{2^{2}}{2^{3}}=\frac{1}{2}$

$\frac{x}{y}=\frac{3^{2}}{3^{3}}=\frac{1}{3}$

$\frac{x}{y}=\frac{4^{2}}{4^{3}}=\frac{1}{4}$

$\frac{x}{y}=\frac{5^{2}}{5^{3}}=\frac{1}{5}$

In all the cases, the ratio is not the same. Therefore, $x$ and $y$ do not vary directly.

Thus, in $(i)$ and $(i i), x$ and $y$ vary directly.