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

$$\begin{gathered} \mathrm{If}x\mathrm{and}y\mathrm{vary}\mathrm{directly},\mathrm{the}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{corresponding}\mathrm{values}\mathrm{of}x\mathrm{and}y\mathrm{remain}s\mathrm{consta}\mathrm{nt}. \\ \left(i\right) \\ \frac{x}{y}=\frac{7}{21}=\frac{1}{3} \\ \frac{x}{y}=\frac{9}{27}=\frac{1}{3} \\ \frac{x}{y}=\frac{13}{39}=\frac{1}{3} \\ \frac{x}{y}=\frac{21}{63}=\frac{1}{3} \\ \frac{\mathrm{x}}{\mathrm{y}}=\frac{25}{75}=\frac{1}{3} \\ \mathrm{In}\mathrm{all}\mathrm{the}\mathrm{cases},\mathrm{the}\mathrm{ratio}\mathrm{is}\mathrm{the}\mathrm{same}.\mathrm{Therefore},x\mathrm{and}y\mathrm{vary}\mathrm{directly}. \\ \left(ii\right) \\ \frac{x}{y}=\frac{10}{5}=2 \\ \frac{x}{y}=\frac{20}{10}=2 \\ \frac{x}{y}=\frac{30}{15}=2 \\ \frac{x}{y}=\frac{40}{20}=2 \\ \frac{x}{y}=\frac{46}{23}=2 \\ \mathrm{In}\mathrm{all}\mathrm{the}\mathrm{cases},\mathrm{the}\mathrm{ratio}\mathrm{is}\mathrm{the}\mathrm{same}.\mathrm{Therefore},x\mathrm{and}y\mathrm{vary}\mathrm{directly}. \\ \left(iii\right) \\ \frac{x}{y}=\frac{2}{6}=\frac{1}{3} \\ \frac{x}{y}=\frac{3}{9}=\frac{1}{3} \\ \frac{x}{y}=\frac{4}{12}=\frac{1}{3} \\ \frac{x}{y}=\frac{5}{17}=\frac{5}{17} \\ \frac{x}{y}=\frac{6}{20}=\frac{3}{10} \\ \mathrm{In}\mathrm{all}\mathrm{the}\mathrm{cases},\mathrm{the}\mathrm{ratio}\mathrm{is}\mathrm{not}\mathrm{the}\mathrm{same}.\mathrm{Therefore},x\mathrm{and}y\mathrm{do}\mathrm{not}\mathrm{vary}\mathrm{directly}. \\ \left(iv\right) \\ \frac{x}{y}=\frac{1^2}{1^3}=1 \\ \frac{x}{y}=\frac{2^2}{2^3}=\frac{1}{2} \\ \frac{x}{y}=\frac{3^2}{3^3}=\frac{1}{3} \\ \frac{x}{y}=\frac{4^2}{4^3}=\frac{1}{4} \\ \frac{x}{y}=\frac{5^2}{5^3}=\frac{1}{5} \\ \mathrm{In}\mathrm{all}\mathrm{the}\mathrm{cases},\mathrm{the}\mathrm{ratio}\mathrm{is}\mathrm{not}\mathrm{the}\mathrm{same}.\mathrm{Therefore},x\mathrm{and}y\mathrm{do}\mathrm{not}\mathrm{vary}\mathrm{directly}. \\ \mathrm{Thus},\mathrm{in}\left(i\right)and\left(ii\right),xandy\mathrm{vary}\mathrm{directly}. \end{gathered}$$