What is 10.213846384638463846... as a Fraction?

10.213846 equals 13617113332 as a fraction.

How do you turn 10.213846 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.213846 repeating into a fraction, begin writing this simple equation:

n = 10.213846 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (3846) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 10213846.3846 (equation 2)

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

1000000 × n = 10213846.3846
    100 × n = 10.213846
 999900 × n = 10212825

10212825999900 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 75 (the GCF - greatest common factor).

n = 10212825999900 = 10212825 ÷ 75999900 ÷ 75 = 13617113332. So,

10.213846 = 13617113332 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 13617113332 is also equal to 10285113332 when expressed as a mixed number.

The repeating decimal 10.213846 (vinculum notation) has a repeated block length of 4. It is also represented as 10.21384638463846... (ellipsis notation) which equals approximately 10.2138463846384638463846 (decimal approximation)(*).

The recurring decimal 10.213846 can be written as a ratio of two integers having 136171 as the numerator and 13332 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.