What is 4.20285285285... as a Fraction?

4.20285 equals 279916660 as a fraction.

How do you turn 4.20285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.20285 repeating into a fraction, begin writing this simple equation:

n = 4.20285 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (285) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁵ = 100000.

100000 × n = 420285.285 (equation 2)

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

100000 × n = 420285.285
   100 × n = 4.20285
 99900 × n = 419865

41986599900 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 15 (the GCF - greatest common factor).

n = 41986599900 = 419865 ÷ 1599900 ÷ 15 = 279916660. So,

4.20285 = 279916660 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 279916660 is also equal to 413516660 when expressed as a mixed number.

The repeating decimal 4.20285 (vinculum notation) has a repeated block length of 3. It is also represented as 4.20285285285... (ellipsis notation) which equals approximately 4.20285285285285285 (decimal approximation)(*).

The recurring decimal 4.20285 can be written as a ratio of two integers having 27991 as the numerator and 6660 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.