What is 4.204285428542854285... as a Fraction?

4.204285 equals 840773199980 as a fraction.

How do you turn 4.204285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.204285 repeating into a fraction, begin writing this simple equation:

n = 4.204285 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (4285) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 4204285.4285 (equation 2)

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

1000000 × n = 4204285.4285
    100 × n = 4.204285
 999900 × n = 4203865

4203865999900 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 5 (the GCF - greatest common factor).

n = 4203865999900 = 4203865 ÷ 5999900 ÷ 5 = 840773199980. So,

4.204285 = 840773199980 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 840773199980 is also equal to 440853199980 when expressed as a mixed number.

The repeating decimal 4.204285 (vinculum notation) has a repeated block length of 4. It is also represented as 4.20428542854285... (ellipsis notation) which equals approximately 4.2042854285428542854285 (decimal approximation)(*).

The recurring decimal 4.204285 can be written as a ratio of two integers having 840773 as the numerator and 199980 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.