What is 15.201515... as a Fraction?

15.2015 equals 10033660 as a fraction.

How do you turn 15.2015 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.2015 repeating into a fraction, begin writing this simple equation:

n = 15.2015 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (15) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁴ = 10000.

10000 × n = 152015.15 (equation 2)

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

10000 × n = 152015.15
  100 × n = 15.2015
 9900 × n = 150495

1504959900 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 15 (the GCF - greatest common factor).

n = 1504959900 = 150495 ÷ 159900 ÷ 15 = 10033660. So,

15.2015 = 10033660 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 10033660 is also equal to 15133660 when expressed as a mixed number.

The repeating decimal 15.2015 (vinculum notation) has a repeated block length of 2. It is also represented as 15.20151515... (ellipsis notation) which equals approximately 15.201515151515 (decimal approximation)(*).

The recurring decimal 15.2015 can be written as a ratio of two integers having 10033 as the numerator and 660 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.