What is 23.10285285285... as a Fraction?

23.10285 equals 307731332 as a fraction.

How do you turn 23.10285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 23.10285 repeating into a fraction, begin writing this simple equation:

n = 23.10285 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (285) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁵ = 100000.

100000 × n = 2310285.285 (equation 2)

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

100000 × n = 2310285.285
   100 × n = 23.10285
 99900 × n = 2307975

230797599900 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 75 (the GCF - greatest common factor).

n = 230797599900 = 2307975 ÷ 7599900 ÷ 75 = 307731332. So,

23.10285 = 307731332 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 307731332 is also equal to 231371332 when expressed as a mixed number.

The repeating decimal 23.10285 (vinculum notation) has a repeated block length of 3. It is also represented as 23.10285285285... (ellipsis notation) which equals approximately 23.10285285285285285 (decimal approximation)(*).

The recurring decimal 23.10285 can be written as a ratio of two integers having 30773 as the numerator and 1332 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.