What is 13.124222... as a Fraction?

13.1242 equals 590594500 as a fraction.

How do you turn 13.1242 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1242 repeating into a fraction, begin writing this simple equation:

n = 13.1242 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 3 digits in the non-repeating part (124). Multiply both sides by 10⁴ = 10000.

10000 × n = 131242.2 (equation 2)

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

10000 × n = 131242.2
 1000 × n = 13.1242
 9000 × n = 118118

1181189000 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 2 (the GCF - greatest common factor).

n = 1181189000 = 118118 ÷ 29000 ÷ 2 = 590594500. So,

13.1242 = 590594500 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 590594500 is also equal to 135594500 when expressed as a mixed number.

The repeating decimal 13.1242 (vinculum notation) has a repeated block length of 1. It is also represented as 13.124222... (ellipsis notation) which equals approximately 13.12422222 (decimal approximation)(*).

The recurring decimal 13.1242 can be written as a ratio of two integers having 59059 as the numerator and 4500 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.