What is 20.9153153153... as a Fraction?

20.9153 equals 11608555 as a fraction.

How do you turn 20.9153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.9153 repeating into a fraction, begin writing this simple equation:

n = 20.9153 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (153) and 1 digit in the non-repeating part (9). Multiply both sides by 10⁴ = 10000.

10000 × n = 209153.153 (equation 2)

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

10000 × n = 209153.153
   10 × n = 20.9153
 9990 × n = 208944

2089449990 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 18 (the GCF - greatest common factor).

n = 2089449990 = 208944 ÷ 189990 ÷ 18 = 11608555. So,

20.9153 = 11608555 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 11608555 is also equal to 20508555 when expressed as a mixed number.

The repeating decimal 20.9153 (vinculum notation) has a repeated block length of 3. It is also represented as 20.9153153153... (ellipsis notation) which equals approximately 20.9153153153153153 (decimal approximation)(*).

The recurring decimal 20.9153 can be written as a ratio of two integers having 11608 as the numerator and 555 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.