​ Write the following rational numbers in p/q form. i. 0. ii. 0. bar 37 iii. 3. bar 17 iv. 15. bar 89 v. 2. bar 514 ​

Question

Write the following rational numbers in p/q form.

i. 0.\(\dot {6}\)

ii. 0. bar 37

iii. 3. bar 17

iv. 15. bar 89

v. 2. bar 514

Answer 1

i. Let x = 0.\(\dot {6}\) …(i) 

∴ x = 0.666… 

Since, one number i.e. 6 is repeating after the decimal point. 

Thus, multiplying both sides by 10, 

10x = 6.666… 

∴ 10 x 6.6 …(ii) 

Subtracting (i) from (ii), 

10x – x = 6.6 – 0.6 

∴ 9x = 6

ii. Let x = 0. bar 37

∴ x = 0.3737… 

Since, two numbers i.e. 3 and 7 are repeating after the decimal point.

Thus, multiplying both sides by 100, 

100x = 37.3737…… 

∴ 100x = 37. bar 37 ……(ii) 

Subtracting (i) from (ii), 

100x – x = 37. bar 37 – 0. bar 37 

∴ 99x = 37 

iii. Let x = 3. bar 17 …(i) 

∴ x = 3.1717… 

Since, two numbers i.e. 1 and 7 are repeating after the decimal point. 

Thus, multiplying both sides by 100, 

100x = 317.1717… 

∴ 100x =  317.17 …(ii) 

Subtracting (i) from (ii), 

100x – x = 317. bar 17 – 3. bar 17 

∴ 99x = 314

iv. Let x = 15. bar 89 …….. (i) 

∴ x = 15.8989… 

Since, two numbers i.e. 8 and 9 are repeating after the decimal point. 

Thus, multiplying both sides by 100, 

100x = 1589.8989… 

∴ 100x = 1589. bar 89 …(ii) 

Subtracting (i) from (ii), 

100x – x = 1589. bar 89 – 15. bar 89

∴ 99x = 1574 

v. Let x = 2. bar 514

∴ x = 2.514514… 

Since, three numbers i.e. 5, 1 and 4 are repeating after the decimal point. 

Thus, multiplying both sides by 1000, 

1000x = 2514.514514… 

1000x = 2514. bar 514 ….(ii) 

Subtracting (i) from (ii), 

1000x – x = 2514. bar 514 – 2. bar 514 

∴ 999x = 2512