Compare 0.1 and 0.111..., 0.13 and 0.131313... And 0.157 and 0.157157157... When written as fractions. Make a conjecture about expressing repeating decimals like these as fractions

Question

Compare 0.1 and 0.111..., 0.13 and 0.131313... And 0.157 and 0.157157157... When written as fractions. Make a conjecture about expressing repeating decimals like these as fractions

Answer 1

1 terminating digit... you divide by 10 ----------> EX. 0.1 = 1/10

1 repeating digit.... you divide that digit by 9 -----> Ex 0.11111.... 0.1 = 1/9

2 terminating digits.... you divide by 100 ----> ex. 0.13 = 13/100

2 repeating digit pattern... you divide by 99 ----> 0.13131313.... = 0.13 = 13/99

3 terminating digits.... you divide by 1000 ----> ex. 0.157 = 157/1000

3 repeating digit pattern... you divide by 999 ----> 0.157157157157.... = 0.157 = 157/999

Use the following table

TERMINATING DECIMAL REPEATING DECIMAL

1 digit divide by 10 divide by 9

2 digit divide by 100 divide by 99

3 digit divide by 1000 divide by 999

4 digit divide by 10000 divide by 9999

etc etc etc

So then what happens with 5 digits? 6 digits?

0.12345 =

0.12345 = 0.123451234512345....

0.123456 =

0.123456123456123456..... = 0.123456