What is 10.10153153153... as a Fraction?

10.10153 equals 11212711100 as a fraction.

How do you turn 10.10153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.10153 repeating into a fraction, begin writing this simple equation:

n = 10.10153 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (153) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁵ = 100000.

100000 × n = 1010153.153 (equation 2)

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

100000 × n = 1010153.153
   100 × n = 10.10153
 99900 × n = 1009143

100914399900 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 = 100914399900 = 1009143 ÷ 999900 ÷ 9 = 11212711100. So,

10.10153 = 11212711100 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 11212711100 is also equal to 10112711100 when expressed as a mixed number.

The repeating decimal 10.10153 (vinculum notation) has a repeated block length of 3. It is also represented as 10.10153153153... (ellipsis notation) which equals approximately 10.10153153153153153 (decimal approximation)(*).

The recurring decimal 10.10153 can be written as a ratio of two integers having 112127 as the numerator and 11100 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.