What is 13.15757... as a Fraction?

13.157 equals 2171165 as a fraction.

How do you turn 13.157 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.157 repeating into a fraction, begin writing this simple equation:

n = 13.157 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (57) and 1 digit in the non-repeating part (1). Multiply both sides by 10³ = 1000.

1000 × n = 13157.57 (equation 2)

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

1000 × n = 13157.57
  10 × n = 13.157
 990 × n = 13026

13026990 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 6 (the GCF - greatest common factor).

n = 13026990 = 13026 ÷ 6990 ÷ 6 = 2171165. So,

13.157 = 2171165 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 2171165 is also equal to 1326165 when expressed as a mixed number.

The repeating decimal 13.157 (vinculum notation) has a repeated block length of 2. It is also represented as 13.1575757... (ellipsis notation) which equals approximately 13.15757575757 (decimal approximation)(*).

The recurring decimal 13.157 can be written as a ratio of two integers having 2171 as the numerator and 165 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.