What is 10.1099... as a Fraction?

10.109 equals 1011100 as a fraction.

How do you turn 10.109 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.109 repeating into a fraction, begin writing this simple equation:

n = 10.109 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 10109.9 (equation 2)

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

1000 × n = 10109.9
 100 × n = 10.109
 900 × n = 9099

9099900 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 9 (the GCF - greatest common factor).

n = 9099900 = 9099 ÷ 9900 ÷ 9 = 1011100. So,

10.109 = 1011100 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 1011100 is also equal to 1011100 when expressed as a mixed number.

The repeating decimal 10.109 (vinculum notation) has a repeated block length of 1. It is also represented as 10.10999... (ellipsis notation) which equals approximately 10.1099999 (decimal approximation)(*).

The recurring decimal 10.109 can be written as a ratio of two integers having 1011 as the numerator and 100 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.