Express the following decimal expression into rational numbers. (i) 0.bar 24 (ii) 2.bar 327 (iii) -5.132 (iv) 3.1 bar 7 (v) 17. bar 215

Question

Express the following decimal expression into rational numbers.

(i) \(0.\overline{24}\)

(ii) \(2.\overline{327}\)

(iii) -5.132

(iv) \(3.1\bar{7}\)

(v) \(17.\overline{215}\)

(vi) \(-21.213\bar{7}\)

Answer 1

(i) \(0.\overline{24}\)

Let x = \(0.\overline{24}\) = 0.24242424 … (1)

(Here period of decimal is 2, multiply equation (1) by 100)

100x = 24.242424 … (2)

(2) – (1)

100x – x = 24.242424… – 0.242424…

99x = 24

x = \(\frac{24}{99}\)

(ii) \(2.\overline{327}\)

Let x = 2.327327327… (1) 

(Here period of decimal is 3, multiply equation (1) by 1000) 

1000x = 2327.327… (2) 

(2) – (1) 

1000x – x = 2327.327327… – 2.327327… 

999x = 2325 

x = \(\frac{2325}{999}\)

(iii) -5.132

x = -5.132 = \(\frac{-5132}{1000}=\frac{-1283}{250}\)

(iv) \(3.1\bar{7}\)

Let x = 3.1777 … (1) 

(Here the repeating decimal digit is 7, which is the second digit after the decimal point, multiply equation (1) by 10) 

10x = 31.7777 … (2) 

(Now period of decimal is 1, multiply equation (2) by 10) 

100x = 317.7777… (3) 

(3) – (2) 

100x – 10x = 317.777… – 31.777…

90x = 286

x = \(\frac{286}{90}=\frac{143}{45}\)

(v) \(17.\overline{215}\)

Let x = 17.215215 … (1) 

1000x = 17215.215215 … (2)

(2) – (1) 

1000x – x = 17215.215215… – 17.215… 

999x = 17198 

x = \(\frac{17198}{999}\)

(vi) \(-21.213\bar{7}\)

Let x = -21.2137777… (1) 

10x = -212.137777… (2) 

100x = -2121.37777… (3) 

1000x = -21213.77777… (4) 

10000x = 212137.77777… (5) 

(Now period of decimal is 1, multiply equation (4) it by 10) 

(5) – (4) 

10000x – 1000x = (-212137.7777…) – (-21213.7777…) 

9000x = -190924

x = \(-\frac{190924}{9000}\)