Question

What are the repeating digits in a small scoop of repeating digits in the decimal equivalent of the fraction shown below 1111/999

Answer 1

Let x = 1111/999. Then

999x = 1111

1000x - x = 1111

Clearly 1 < x < 2, and x must have a repeating pattern of 3 digits immediately after the decimal point. So we can write

x = 1.abcabcabc…

Then

1000x = 1abc.abcabcabc

and

1000x - x = 1abc.abcabcabc… - 1.abcabcabc…

999x = 1abc - 1

x = (1abc - 1)/999

Now,

1abc - 1 = 1000 + 100a + 10b + c - 1

and

1111 = 1000 + 100 + 10 + 1

which means a = b = 1 and c = 2.

So we have

1111/999 = 1.112112112…