What is 21.11153153153... as a Fraction?

21.11153 equals 1171695550 as a fraction.

How do you turn 21.11153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.11153 repeating into a fraction, begin writing this simple equation:

n = 21.11153 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (153) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁵ = 100000.

100000 × n = 2111153.153 (equation 2)

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

100000 × n = 2111153.153
   100 × n = 21.11153
 99900 × n = 2109042

210904299900 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 = 210904299900 = 2109042 ÷ 1899900 ÷ 18 = 1171695550. So,

21.11153 = 1171695550 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 1171695550 is also equal to 216195550 when expressed as a mixed number.

The repeating decimal 21.11153 (vinculum notation) has a repeated block length of 3. It is also represented as 21.11153153153... (ellipsis notation) which equals approximately 21.11153153153153153 (decimal approximation)(*).

The recurring decimal 21.11153 can be written as a ratio of two integers having 117169 as the numerator and 5550 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.