Question

Convert the following decimals into p/q form

(1) 0.5555.........

​​​​(2) 0.151515..........

(3) 0.78888.................

(4) 31.2999......................

(5) 63.95.............................

Answer 1

In option (2) I am going to try to explain using the repeating decimal 0.15151515... Write 0.15151515... as 15/100 + 15/10000 + 15/1000000 + 15/100000000 + ... =15/100 + 15/1002 + 15/1003 + 15/1004 + ... This is the geometric series a + ar + ar2 + ar3 + ar4 + ... with a = 15/100 and r = 1/100. |r| < 1 then the series a + ar + ar2 + ar3 + ar4 + ... approaches a/(1 - r) as the number of terms approaches infinity. In the case of 0.15151515.. this means that 15/100 + 15/10000 + 15/1000000 + 15/100000000 + ... =15/100 + 15/1002 + 15/1003 + 15/1004 + ... =(15/100)/(1 -1/100) = 0.15/99 =(15/100)/(99/100) =15/99 similarly in option(1), option (3),and option (4) and option(5) we can do the same