What is 2.20428571428571428571428571428571428571... as a Fraction?

2.20428571 equals 1543700 as a fraction.

How do you turn 2.20428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.20428571 repeating into a fraction, begin writing this simple equation:

n = 2.20428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 220428571.428571 (equation 2)

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

100000000 × n = 220428571.428571
      100 × n = 2.20428571
 99999900 × n = 220428351

22042835199999900 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 142857 (the GCF - greatest common factor).

n = 22042835199999900 = 220428351 ÷ 14285799999900 ÷ 142857 = 1543700. So,

2.20428571 = 1543700 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 1543700 is also equal to 2143700 when expressed as a mixed number.

The repeating decimal 2.20428571 (vinculum notation) has a repeated block length of 6. It is also represented as 2.20428571428571428571... (ellipsis notation) which equals approximately 2.20428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 2.20428571 can be written as a ratio of two integers having 1543 as the numerator and 700 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.