How does 0.4 (repeating) equal 4/9?

Question

Following the steps to convert decimals into fractions, I tried 4/10, 44/100, and 444/1000 and got 2/5, 11/25, and 111/250 respectively, all apparently incorrect. I’m not sure how to handle repeating decimals in this scenario.

Answer 1

When the single decimal digit repeats, you divide that digit by 9, not 10.

0.444444.... = 0.4 = 4/9

0.77777777= 0.7 = 7/9

With TWO repeating decimals, you divide by 99, not 100...

EX.

0.2323232323.... 0.23=23/99

With Three repeating decimals, you divide by 999, not 1000...

EX. 0.123123123.... = 0.123 = 123/999

etc

The following table should help you change the decimals

by dividing by the number shown in the table...

# of digits repeating Does not repeat

1 9 10

2 99 100

3 999 1000

4 9999 10000

etc etc etc

EX. 0.123412341234123412341234...... = 0.1234 = 1234/9999