What is 10.108666... as a Fraction?

10.1086 equals 151631500 as a fraction.

How do you turn 10.1086 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1086 repeating into a fraction, begin writing this simple equation:

n = 10.1086 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 3 digits in the non-repeating part (108). Multiply both sides by 10⁴ = 10000.

10000 × n = 101086.6 (equation 2)

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

10000 × n = 101086.6
 1000 × n = 10.1086
 9000 × n = 90978

909789000 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 = 909789000 = 90978 ÷ 69000 ÷ 6 = 151631500. So,

10.1086 = 151631500 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 151631500 is also equal to 101631500 when expressed as a mixed number.

The repeating decimal 10.1086 (vinculum notation) has a repeated block length of 1. It is also represented as 10.108666... (ellipsis notation) which equals approximately 10.10866666 (decimal approximation)(*).

The recurring decimal 10.1086 can be written as a ratio of two integers having 15163 as the numerator and 1500 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.