What is 5.157142714271427142... as a Fraction?

5.157142 equals 5156627999900 as a fraction.

How do you turn 5.157142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.157142 repeating into a fraction, begin writing this simple equation:

n = 5.157142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142) and 2 digits in the non-repeating part (15). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 5157142.7142 (equation 2)

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

1000000 × n = 5157142.7142
    100 × n = 5.157142
 999900 × n = 5156627

n = 5156627999900 (answer)

5.157142 = 5156627999900 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 5156627999900 is also equal to 5157127999900 when expressed as a mixed number.

The repeating decimal 5.157142 (vinculum notation) has a repeated block length of 4. It is also represented as 5.15714271427142... (ellipsis notation) which equals approximately 5.1571427142714271427142 (decimal approximation)(*).

The recurring decimal 5.157142 can be written as a ratio of two integers having 5156627 as the numerator and 999900 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.