What is 100.11111... as a Fraction?

100.111 equals 9019 as a fraction.

How do you turn 100.111 repeating into a fraction?

Detailed Answer:

Step 1: To convert 100.111 repeating into a fraction, begin writing this simple equation:

n = 100.111 (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 = 100111.1 (equation 2)

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

1000 × n = 100111.1
 100 × n = 100.111
 900 × n = 90100

90100900 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 = 90100900 = 90100 ÷ 100900 ÷ 100 = 9019. So,

100.111 = 9019 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 9019 is also equal to 10019 when expressed as a mixed number.

The repeating decimal 100.111 (vinculum notation) has a repeated block length of 1. It is also represented as 100.11111... (ellipsis notation) which equals approximately 100.1111111 (decimal approximation)(*).

The recurring decimal 100.111 can be written as a ratio of two integers having 901 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.