What is 4.21285285285... as a Fraction?

4.21285 equals 350728325 as a fraction.

How do you turn 4.21285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.21285 repeating into a fraction, begin writing this simple equation:

n = 4.21285 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (285) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁵ = 100000.

100000 × n = 421285.285 (equation 2)

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

100000 × n = 421285.285
   100 × n = 4.21285
 99900 × n = 420864

42086499900 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 12 (the GCF - greatest common factor).

n = 42086499900 = 420864 ÷ 1299900 ÷ 12 = 350728325. So,

4.21285 = 350728325 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 350728325 is also equal to 417728325 when expressed as a mixed number.

The repeating decimal 4.21285 (vinculum notation) has a repeated block length of 3. It is also represented as 4.21285285285... (ellipsis notation) which equals approximately 4.21285285285285285 (decimal approximation)(*).

The recurring decimal 4.21285 can be written as a ratio of two integers having 35072 as the numerator and 8325 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.