What is 11.112222... as a Fraction?

11.1122 equals 10001900 as a fraction.

How do you turn 11.1122 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1122 repeating into a fraction, begin writing this simple equation:

n = 11.1122 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 3 digits in the non-repeating part (112). Multiply both sides by 10⁴ = 10000.

10000 × n = 111122.2 (equation 2)

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

10000 × n = 111122.2
 1000 × n = 11.1122
 9000 × n = 100010

1000109000 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 10 (the GCF - greatest common factor).

n = 1000109000 = 100010 ÷ 109000 ÷ 10 = 10001900. So,

11.1122 = 10001900 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 10001900 is also equal to 11101900 when expressed as a mixed number.

The repeating decimal 11.1122 (vinculum notation) has a repeated block length of 1. It is also represented as 11.112222... (ellipsis notation) which equals approximately 11.11222222 (decimal approximation)(*).

The recurring decimal 11.1122 can be written as a ratio of two integers having 10001 as the numerator and 900 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.