What is 10.122222... as a Fraction?

10.1222 equals 91190 as a fraction.

How do you turn 10.1222 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1222 repeating into a fraction, begin writing this simple equation:

n = 10.1222 (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 = 101222.2 (equation 2)

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

10000 × n = 101222.2
 1000 × n = 10.1222
 9000 × n = 91100

911009000 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 = 911009000 = 91100 ÷ 1009000 ÷ 100 = 91190. So,

10.1222 = 91190 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 91190 is also equal to 101190 when expressed as a mixed number.

The repeating decimal 10.1222 (vinculum notation) has a repeated block length of 1. It is also represented as 10.122222... (ellipsis notation) which equals approximately 10.12222222 (decimal approximation)(*).

The recurring decimal 10.1222 can be written as a ratio of two integers having 911 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.