#### 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 |
1^{2} |
2^{2} |
3^{2} |
4^{2} |
5^{2} |

b |
1^{3} |
2^{3} |
3^{3} |
4^{3} |
5^{3} |

#### Solution

\[\text{ If x and y vary directly, the ratio of the corresponding values of x and y remains constant} . \]

\[(i)\]

\[\frac{x}{y} = \frac{7}{21} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{9}{27} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{13}{39} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{21}{63} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{25}{75} = \frac{1}{3}\]

\[\text{ In all the cases, the ratio is the same . Therefore, x and y vary directly } . \]

\[(ii)\]

\[\frac{x}{y} = \frac{10}{5} = 2\]

\[\frac{x}{y} = \frac{20}{10} = 2\]

\[\frac{x}{y} = \frac{30}{15} = 2\]

\[\frac{x}{y} = \frac{40}{20} = 2\]

\[\frac{x}{y} = \frac{46}{23} = 2\]

\[\text{ In all the cases, the ratio is the same . Therefore, x and y vary directly . } \]

\[(iii)\]

\[\frac{x}{y} = \frac{2}{6} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{3}{9} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{4}{12} = \frac{1}{3}\]

\[\frac{x}{y} = \frac{5}{17} = \frac{5}{17}\]

\[\frac{x}{y} = \frac{6}{20} = \frac{3}{10}\]

\[\text{ In all the cases, the ratio is not the same . Therefore, x and y do not vary directly } . \]

\[(iv)\]

\[\frac{x}{y} = \frac{1^2}{1^3} = 1\]

\[\frac{x}{y} = \frac{2^2}{2^3} = \frac{1}{2}\]

\[\frac{x}{y} = \frac{3^2}{3^3} = \frac{1}{3}\]

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

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

\[\text{ In all the cases, the ratio is not the same . Therefore, x and y do not vary directly . } \]

\[\text{ Thus, in (i) and (ii), x and y vary directly .} \]