What is 5.181538153815381538... as a Fraction?

5.181538 equals 25905149995 as a fraction.

How do you turn 5.181538 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.181538 repeating into a fraction, begin writing this simple equation:

n = 5.181538 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1538) and 2 digits in the non-repeating part (18). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 5181538.1538 (equation 2)

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

1000000 × n = 5181538.1538
    100 × n = 5.181538
 999900 × n = 5181020

5181020999900 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 20 (the GCF - greatest common factor).

n = 5181020999900 = 5181020 ÷ 20999900 ÷ 20 = 25905149995. So,

5.181538 = 25905149995 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 25905149995 is also equal to 5907649995 when expressed as a mixed number.

The repeating decimal 5.181538 (vinculum notation) has a repeated block length of 4. It is also represented as 5.18153815381538... (ellipsis notation) which equals approximately 5.1815381538153815381538 (decimal approximation)(*).

The recurring decimal 5.181538 can be written as a ratio of two integers having 259051 as the numerator and 49995 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.