What is 3.10714714714... as a Fraction?

3.10714 equals 258678325 as a fraction.

How do you turn 3.10714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.10714 repeating into a fraction, begin writing this simple equation:

n = 3.10714 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (714) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁵ = 100000.

100000 × n = 310714.714 (equation 2)

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

100000 × n = 310714.714
   100 × n = 3.10714
 99900 × n = 310404

31040499900 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 12 (the GCF - greatest common factor).

n = 31040499900 = 310404 ÷ 1299900 ÷ 12 = 258678325. So,

3.10714 = 258678325 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 258678325 is also equal to 38928325 when expressed as a mixed number.

The repeating decimal 3.10714 (vinculum notation) has a repeated block length of 3. It is also represented as 3.10714714714... (ellipsis notation) which equals approximately 3.10714714714714714 (decimal approximation)(*).

The recurring decimal 3.10714 can be written as a ratio of two integers having 25867 as the numerator and 8325 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.