What is 2.1909090909090... as a Fraction?

2.190 equals 241110 as a fraction.

How do you turn 2.190 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.190 repeating into a fraction, begin writing this simple equation:

n = 2.190 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (90) and 1 digit in the non-repeating part (1). Multiply both sides by 10³ = 1000.

1000 × n = 2190.90 (equation 2)

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

1000 × n = 2190.90
  10 × n = 2.190
 990 × n = 2169

2169990 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 = 2169990 = 2169 ÷ 9990 ÷ 9 = 241110. So,

2.190 = 241110 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 241110 is also equal to 221110 when expressed as a mixed number.

The repeating decimal 2.190 (vinculum notation) has a repeated block length of 2. It is also represented as 2.1909090... (ellipsis notation) which equals approximately 2.19090909090 (decimal approximation)(*).

The recurring decimal 2.190 can be written as a ratio of two integers having 241 as the numerator and 110 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.