What is 1.11111... as a Fraction?

1.111 equals 109 as a fraction.

How do you turn 1.111 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.111 repeating into a fraction, begin writing this simple equation:

n = 1.111 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (1) and 2 digits in the non-repeating part (11). Multiply both sides by 10³ = 1000.

1000 × n = 1111.1 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

1000 × n = 1111.1
 100 × n = 1.111
 900 × n = 1000

1000900 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 100 (the GCF - greatest common factor).

n = 1000900 = 1000 ÷ 100900 ÷ 100 = 109. So,

1.111 = 109 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 109 is also equal to 119 when expressed as a mixed number.

The repeating decimal 1.111 (vinculum notation) has a repeated block length of 1. It is also represented as 1.11111... (ellipsis notation) which equals approximately 1.1111111 (decimal approximation)(*).

The recurring decimal 1.111 can be written as a ratio of two integers having 10 as the numerator and 9 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.