What is 5.1714714714... as a Fraction?

5.1714 equals 172213330 as a fraction.

How do you turn 5.1714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.1714 repeating into a fraction, begin writing this simple equation:

n = 5.1714 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (714) and 1 digit in the non-repeating part (1). Multiply both sides by 10⁴ = 10000.

10000 × n = 51714.714 (equation 2)

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

10000 × n = 51714.714
   10 × n = 5.1714
 9990 × n = 51663

516639990 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 = 516639990 = 51663 ÷ 39990 ÷ 3 = 172213330. So,

5.1714 = 172213330 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 172213330 is also equal to 55713330 when expressed as a mixed number.

The repeating decimal 5.1714 (vinculum notation) has a repeated block length of 3. It is also represented as 5.1714714714... (ellipsis notation) which equals approximately 5.1714714714714714 (decimal approximation)(*).

The recurring decimal 5.1714 can be written as a ratio of two integers having 17221 as the numerator and 3330 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.