What is 10.11818... as a Fraction?

10.118 equals 1113110 as a fraction.

How do you turn 10.118 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.118 repeating into a fraction, begin writing this simple equation:

n = 10.118 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (18) and 1 digit in the non-repeating part (1). Multiply both sides by 10³ = 1000.

1000 × n = 10118.18 (equation 2)

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

1000 × n = 10118.18
  10 × n = 10.118
 990 × n = 10017

10017990 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 9 (the GCF - greatest common factor).

n = 10017990 = 10017 ÷ 9990 ÷ 9 = 1113110. So,

10.118 = 1113110 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 1113110 is also equal to 1013110 when expressed as a mixed number.

The repeating decimal 10.118 (vinculum notation) has a repeated block length of 2. It is also represented as 10.1181818... (ellipsis notation) which equals approximately 10.11818181818 (decimal approximation)(*).

The recurring decimal 10.118 can be written as a ratio of two integers having 1113 as the numerator and 110 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.