Question

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

(i)

a	7	9	13	21	25
b	21	27	39	63	75

(ii)

a	10	20	30	40	46
b	5	10	15	20	23

(iii)

a	2	3	4	5	6
b	6	9	12	17	20

(iv)

a	12	22	32	42	52
b	13	23	33	43	53

Answer 1

\[\text{ If x and y vary directly, the ratio of the corresponding values of x and y remains constant}  . \]
\[(i)\]
\[\frac{x}{y} = \frac{7}{21} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{9}{27} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{13}{39} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{21}{63} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{25}{75} = \frac{1}{3}\]
\[\text{ In all the cases, the ratio is the same . Therefore, x and y vary directly } . \]
\[(ii)\]
\[\frac{x}{y} = \frac{10}{5} = 2\]
\[\frac{x}{y} = \frac{20}{10} = 2\]
\[\frac{x}{y} = \frac{30}{15} = 2\]
\[\frac{x}{y} = \frac{40}{20} = 2\]
\[\frac{x}{y} = \frac{46}{23} = 2\]
\[\text{ In all the cases, the ratio is the same . Therefore, x and y vary directly . } \]
\[(iii)\]
\[\frac{x}{y} = \frac{2}{6} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{3}{9} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{4}{12} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{5}{17} = \frac{5}{17}\]
\[\frac{x}{y} = \frac{6}{20} = \frac{3}{10}\]
\[\text{ In all the cases, the ratio is not the same . Therefore, x and y do not vary directly } . \]
\[(iv)\]
\[\frac{x}{y} = \frac{1^2}{1^3} = 1\]
\[\frac{x}{y} = \frac{2^2}{2^3} = \frac{1}{2}\]
\[\frac{x}{y} = \frac{3^2}{3^3} = \frac{1}{3}\]
\[\frac{x}{y} = \frac{4^2}{4^3} = \frac{1}{4}\]
\[\frac{x}{y} = $\frac{5^2}{5^3}$ = \frac{1}{5}\]
\[\text{ In all the cases, the ratio is not the same . Therefore, x and y do not vary directly . } \]
\[\text{ Thus, in (i) and (ii), x and y vary directly .} \]