What is 0.214545... as a Fraction?

0.2145 equals 59275 as a fraction.

How do you turn 0.2145 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.2145 repeating into a fraction, begin writing this simple equation:

n = 0.2145 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (45) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 2145.45 (equation 2)

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

10000 × n = 2145.45
  100 × n = 0.2145
 9900 × n = 2124

21249900 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 36 (the GCF - greatest common factor).

n = 21249900 = 2124 ÷ 369900 ÷ 36 = 59275. So,

0.2145 = 59275 as the lowest possible fraction.

The repeating decimal 0.2145 (vinculum notation) has a repeated block length of 2. It is also represented as 0.21454545... (ellipsis notation) which equals approximately 0.214545454545 (decimal approximation)(*).

The recurring decimal 0.2145 can be written as a ratio of two integers having 59 as the numerator and 275 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.