What is 0.215757... as a Fraction?

0.2157 equals 178825 as a fraction.

How do you turn 0.2157 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.2157 repeating into a fraction, begin writing this simple equation:

n = 0.2157 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (57) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 2157.57 (equation 2)

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

10000 × n = 2157.57
  100 × n = 0.2157
 9900 × n = 2136

21369900 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 12 (the GCF - greatest common factor).

n = 21369900 = 2136 ÷ 129900 ÷ 12 = 178825. So,

0.2157 = 178825 as the lowest possible fraction.

The repeating decimal 0.2157 (vinculum notation) has a repeated block length of 2. It is also represented as 0.21575757... (ellipsis notation) which equals approximately 0.215757575757 (decimal approximation)(*).

The recurring decimal 0.2157 can be written as a ratio of two integers having 178 as the numerator and 825 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.