What is 1.1009090909... as a Fraction?

1.1009 equals 12111100 as a fraction.

How do you turn 1.1009 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.1009 repeating into a fraction, begin writing this simple equation:

n = 1.1009 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (09) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁴ = 10000.

10000 × n = 11009.09 (equation 2)

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

10000 × n = 11009.09
  100 × n = 1.1009
 9900 × n = 10899

108999900 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 = 108999900 = 10899 ÷ 99900 ÷ 9 = 12111100. So,

1.1009 = 12111100 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 12111100 is also equal to 11111100 when expressed as a mixed number.

The repeating decimal 1.1009 (vinculum notation) has a repeated block length of 2. It is also represented as 1.10090909... (ellipsis notation) which equals approximately 1.100909090909 (decimal approximation)(*).

The recurring decimal 1.1009 can be written as a ratio of two integers having 1211 as the numerator and 1100 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.