What is 4.138571485714857148571485714... as a Fraction?

4.1385714 equals 413853019999900 as a fraction.

How do you turn 4.1385714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.1385714 repeating into a fraction, begin writing this simple equation:

n = 4.1385714 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (85714) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 41385714.85714 (equation 2)

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

10000000 × n = 41385714.85714
     100 × n = 4.1385714
 9999900 × n = 41385301

n = 413853019999900 (answer)

4.1385714 = 413853019999900 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 413853019999900 is also equal to 413857019999900 when expressed as a mixed number.

The repeating decimal 4.1385714 (vinculum notation) has a repeated block length of 5. It is also represented as 4.13857148571485714... (ellipsis notation) which equals approximately 4.138571485714857148571485714 (decimal approximation)(*).

The recurring decimal 4.1385714 can be written as a ratio of two integers having 41385301 as the numerator and 9999900 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.