What is 85.12666... as a Fraction?

85.126 equals 12769150 as a fraction.

How do you turn 85.126 repeating into a fraction?

Detailed Answer:

Step 1: To convert 85.126 repeating into a fraction, begin writing this simple equation:

n = 85.126 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 85126.6 (equation 2)

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

1000 × n = 85126.6
 100 × n = 85.126
 900 × n = 76614

76614900 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 6 (the GCF - greatest common factor).

n = 76614900 = 76614 ÷ 6900 ÷ 6 = 12769150. So,

85.126 = 12769150 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 12769150 is also equal to 8519150 when expressed as a mixed number.

The repeating decimal 85.126 (vinculum notation) has a repeated block length of 1. It is also represented as 85.12666... (ellipsis notation) which equals approximately 85.1266666 (decimal approximation)(*).

The recurring decimal 85.126 can be written as a ratio of two integers having 12769 as the numerator and 150 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.