What is 10.115454... as a Fraction?

10.1154 equals 111271100 as a fraction.

How do you turn 10.1154 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1154 repeating into a fraction, begin writing this simple equation:

n = 10.1154 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (54) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁴ = 10000.

10000 × n = 101154.54 (equation 2)

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

10000 × n = 101154.54
  100 × n = 10.1154
 9900 × n = 100143

1001439900 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 = 1001439900 = 100143 ÷ 99900 ÷ 9 = 111271100. So,

10.1154 = 111271100 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 111271100 is also equal to 101271100 when expressed as a mixed number.

The repeating decimal 10.1154 (vinculum notation) has a repeated block length of 2. It is also represented as 10.11545454... (ellipsis notation) which equals approximately 10.115454545454 (decimal approximation)(*).

The recurring decimal 10.1154 can be written as a ratio of two integers having 11127 as the numerator and 1100 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.