What is 13.15292929... as a Fraction?

13.1529 equals 651074950 as a fraction.

How do you turn 13.1529 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1529 repeating into a fraction, begin writing this simple equation:

n = 13.1529 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (29) and 2 digits in the non-repeating part (15). Multiply both sides by 10⁴ = 10000.

10000 × n = 131529.29 (equation 2)

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

10000 × n = 131529.29
  100 × n = 13.1529
 9900 × n = 130214

1302149900 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 = 1302149900 = 130214 ÷ 29900 ÷ 2 = 651074950. So,

13.1529 = 651074950 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 651074950 is also equal to 137574950 when expressed as a mixed number.

The repeating decimal 13.1529 (vinculum notation) has a repeated block length of 2. It is also represented as 13.15292929... (ellipsis notation) which equals approximately 13.152929292929 (decimal approximation)(*).

The recurring decimal 13.1529 can be written as a ratio of two integers having 65107 as the numerator and 4950 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.