Express the following numbers in the form , where p and q are both integers and q ≠ 0: (i) 0. ̅3 (ii) 5. ̅2 (iii) 0.404040... (iv) 0.4 ̅7 (v) 0.1 ̅3 ̅4 (vi) 0. ̅0 ̅0 ̅1

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KnowledgeBoat · Accepted Answer

(i) Let x = = 0.333333... As there is one repeating digit after decimal point, So multiplying both sides of (i) by 10 we get, 10x = 3.3333... Subtracting (i) from (ii), we get 9x = 3 x = = , which is in the form of , q ≠ 0. (ii) Let x = = 5.2222... As there is one repeating digit after decimal point, So multiplying both sides of (i) by 10 we get, 10x = 52.2222... Subtracting (i) from (ii), we get 9x = 47 x = , Which is in the form of , q ≠ 0. (iii) Let x = = 0.4040... As there are two repeating digit after decimal point, So multiplying both sides of (i) by 100 we get, 100x = 40.4040... Subtracting (i) from (ii), we get 99x = 40 x = , Which is in the form of , q ≠ 0 (iv) Let x = = 0.477777... As there is one repeating digit after decimal point , So multiplying both sides of (i) by 10 we get, 10x=4.7777... Multiply by 100 on both sides 100x=47.77777..... subtracting (ii) from (iii), we get 100x-10x=47.7777.. -4.777... 90x= 43 x = Which is in the form of , q ≠ 0 (v) Let x = = 0.13434 ... So multiplying both sides of (i) by 10 we get, 10x=1.343434... Again multiply by 100 on both sides , 1000x =134.3434..... Subtracting (ii) from (iii), we get 1000x - 10x = 134.3434... - 1.3434... 990x = 133 x = which is in the form of , q ≠ 0 (vi) Let x = = 0.001001001... So multiplying both sides of (i) by 1000, we get, 1000x = 1.001001... Subtracting (i) from (ii), we get 1000x - x = 1.001001... - 0.001001... 999x = 1 x = which is in the form of , q ≠ 0.

Mathematics

Express the following numbers in the form , where p and q are both integers and q ≠ 0: (i) 0. ̅3 (ii) 5. ̅2 (iii) 0.404040... (iv) 0.4 ̅7 (v) 0.1 ̅3 ̅4 (vi) 0. ̅0 ̅0 ̅1

Rational Irrational Nos

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