What is 1.211428142814281428... as a Fraction?

1.211428 equals 403769333300 as a fraction.

How do you turn 1.211428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.211428 repeating into a fraction, begin writing this simple equation:

n = 1.211428 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1428) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 1211428.1428 (equation 2)

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

1000000 × n = 1211428.1428
    100 × n = 1.211428
 999900 × n = 1211307

1211307999900 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 3 (the GCF - greatest common factor).

n = 1211307999900 = 1211307 ÷ 3999900 ÷ 3 = 403769333300. So,

1.211428 = 403769333300 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 403769333300 is also equal to 170469333300 when expressed as a mixed number.

The repeating decimal 1.211428 (vinculum notation) has a repeated block length of 4. It is also represented as 1.21142814281428... (ellipsis notation) which equals approximately 1.2114281428142814281428 (decimal approximation)(*).

The recurring decimal 1.211428 can be written as a ratio of two integers having 403769 as the numerator and 333300 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.