What is 5.125757... as a Fraction?

5.1257 equals 3383660 as a fraction.

How do you turn 5.1257 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.1257 repeating into a fraction, begin writing this simple equation:

n = 5.1257 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (57) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 51257.57 (equation 2)

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

10000 × n = 51257.57
  100 × n = 5.1257
 9900 × n = 50745

507459900 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 15 (the GCF - greatest common factor).

n = 507459900 = 50745 ÷ 159900 ÷ 15 = 3383660. So,

5.1257 = 3383660 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 3383660 is also equal to 583660 when expressed as a mixed number.

The repeating decimal 5.1257 (vinculum notation) has a repeated block length of 2. It is also represented as 5.12575757... (ellipsis notation) which equals approximately 5.125757575757 (decimal approximation)(*).

The recurring decimal 5.1257 can be written as a ratio of two integers having 3383 as the numerator and 660 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.