What is 11.122222... as a Fraction?

11.1222 equals 100190 as a fraction.

How do you turn 11.1222 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1222 repeating into a fraction, begin writing this simple equation:

n = 11.1222 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 3 digits in the non-repeating part (122). Multiply both sides by 10⁴ = 10000.

10000 × n = 111222.2 (equation 2)

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

10000 × n = 111222.2
 1000 × n = 11.1222
 9000 × n = 100100

1001009000 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 = 1001009000 = 100100 ÷ 1009000 ÷ 100 = 100190. So,

11.1222 = 100190 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 100190 is also equal to 111190 when expressed as a mixed number.

The repeating decimal 11.1222 (vinculum notation) has a repeated block length of 1. It is also represented as 11.122222... (ellipsis notation) which equals approximately 11.12222222 (decimal approximation)(*).

The recurring decimal 11.1222 can be written as a ratio of two integers having 1001 as the numerator and 90 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.