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

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Solution

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

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Similar questions

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

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

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

a	1²	2²	3²	4²	5²
b	1³	2³	3³	4³	5³

Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data:

24, 22, 23, 14, 15, 7, 21

17, 15, 9, 13, 21, 32, 42, 7, 12, 10

$x$	$15$	$16$	$17$	$18$	$19$	$20$	$21$	$22$	$23$
$f$	$6$	$7$	$9$	$13$	$10$	$12$	$8$	$0$	$4$

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