What is 21.3153153153... as a Fraction?

21.3153 equals 2366111 as a fraction.

How do you turn 21.3153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.3153 repeating into a fraction, begin writing this simple equation:

n = 21.3153 (equation 1)

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

10000 × n = 213153.153 (equation 2)

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

10000 × n = 213153.153
   10 × n = 21.3153
 9990 × n = 212940

2129409990 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 90 (the GCF - greatest common factor).

n = 2129409990 = 212940 ÷ 909990 ÷ 90 = 2366111. So,

21.3153 = 2366111 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 2366111 is also equal to 2135111 when expressed as a mixed number.

The repeating decimal 21.3153 (vinculum notation) has a repeated block length of 3. It is also represented as 21.3153153153... (ellipsis notation) which equals approximately 21.3153153153153153 (decimal approximation)(*).

The recurring decimal 21.3153 can be written as a ratio of two integers having 2366 as the numerator and 111 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.