What is 5.17838383838383... as a Fraction?

5.1783 equals 256334950 as a fraction.

How do you turn 5.1783 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.1783 repeating into a fraction, begin writing this simple equation:

n = 5.1783 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (83) and 2 digits in the non-repeating part (17). Multiply both sides by 10⁴ = 10000.

10000 × n = 51783.83 (equation 2)

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

10000 × n = 51783.83
  100 × n = 5.1783
 9900 × n = 51266

512669900 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 2 (the GCF - greatest common factor).

n = 512669900 = 51266 ÷ 29900 ÷ 2 = 256334950. So,

5.1783 = 256334950 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 256334950 is also equal to 58834950 when expressed as a mixed number.

The repeating decimal 5.1783 (vinculum notation) has a repeated block length of 2. It is also represented as 5.17838383... (ellipsis notation) which equals approximately 5.178383838383 (decimal approximation)(*).

The recurring decimal 5.1783 can be written as a ratio of two integers having 25633 as the numerator and 4950 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.