What is 10.10142142142... as a Fraction?

10.10142 equals 25228324975 as a fraction.

How do you turn 10.10142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.10142 repeating into a fraction, begin writing this simple equation:

n = 10.10142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁵ = 100000.

100000 × n = 1010142.142 (equation 2)

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

100000 × n = 1010142.142
   100 × n = 10.10142
 99900 × n = 1009132

100913299900 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 4 (the GCF - greatest common factor).

n = 100913299900 = 1009132 ÷ 499900 ÷ 4 = 25228324975. So,

10.10142 = 25228324975 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 25228324975 is also equal to 10253324975 when expressed as a mixed number.

The repeating decimal 10.10142 (vinculum notation) has a repeated block length of 3. It is also represented as 10.10142142142... (ellipsis notation) which equals approximately 10.10142142142142142 (decimal approximation)(*).

The recurring decimal 10.10142 can be written as a ratio of two integers having 252283 as the numerator and 24975 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.