Question

Express in p /q form (i) 5.1111….(ii) 0.2 (iii) 1.6323232… (iv) 0.404040…..

Answer 1

$i) Let \: x = 5.\bar{1} \: --(1)$

/* Multiplying equation (1) by 10, we get */

$\implies 10x = 51.\bar{1} \: --(2)$

/* Subtract equation (1) from equation (2), we get */

$\implies 9x = 46$

$\implies x = \frac{46}{9} \: \blue{ (\frac{p}{q} \: form )}$

$ii)Given \: decimal \:form = 0.2$

$= 0.2 \times \frac{10}{10}$

$= \frac{2}{10}$

$= \frac{1}{10} \: \blue{ (\frac{p}{q} \: form )}$

$iii) Let \: y = 1.6\bar{32} \: --(4)$

/* Multiplying equation (1) by 10, we get */

$\implies 10x = 16.\bar{32} \: --(5)$

/*Multiplying equation (5) by 100, we get */

$\implies 1000x = 1632.\bar{32} \: --(6)$

/* Subtract equation (5) from equation (6), we get */

$\implies 990x = 1616$

$\implies x = \frac{1616}{990}$

$= \frac{808}{495} \: \blue{ (\frac{p}{q} \: form )}$

$\therefore \red{ 1.6323232\ldots}\green{=\frac{808}{495}}$

$iv) Let \: z = 0.404040 \ldots \:--(7)$

/* Multiplying equation (7) by 100 , we get */

$\implies 100z = 40.\bar{40} \: --(8)$

/* Subtract equation (7) from equation (8), we get */

$\implies 99z = 40$

$\implies z = \frac{40}{99} \: \blue{( \frac{p}{q} \: form )}$

$\therefore \red{ 0.404040 \ldots} \green {= \frac{40}{99}}$

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