What is 20.235454... as a Fraction?

20.2354 equals 222591100 as a fraction.

How do you turn 20.2354 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.2354 repeating into a fraction, begin writing this simple equation:

n = 20.2354 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (54) and 2 digits in the non-repeating part (23). Multiply both sides by 10⁴ = 10000.

10000 × n = 202354.54 (equation 2)

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

10000 × n = 202354.54
  100 × n = 20.2354
 9900 × n = 200331

2003319900 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 = 2003319900 = 200331 ÷ 99900 ÷ 9 = 222591100. So,

20.2354 = 222591100 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 222591100 is also equal to 202591100 when expressed as a mixed number.

The repeating decimal 20.2354 (vinculum notation) has a repeated block length of 2. It is also represented as 20.23545454... (ellipsis notation) which equals approximately 20.235454545454 (decimal approximation)(*).

The recurring decimal 20.2354 can be written as a ratio of two integers having 22259 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.