What is 15.1999... as a Fraction?

15.199 equals 765 as a fraction.

How do you turn 15.199 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.199 repeating into a fraction, begin writing this simple equation:

n = 15.199 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (19). Multiply both sides by 10³ = 1000.

1000 × n = 15199.9 (equation 2)

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

1000 × n = 15199.9
 100 × n = 15.199
 900 × n = 13680

13680900 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 180 (the GCF - greatest common factor).

n = 13680900 = 13680 ÷ 180900 ÷ 180 = 765. So,

15.199 = 765 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 765 is also equal to 1515 when expressed as a mixed number.

The repeating decimal 15.199 (vinculum notation) has a repeated block length of 1. It is also represented as 15.19999... (ellipsis notation) which equals approximately 15.1999999 (decimal approximation)(*).

The recurring decimal 15.199 can be written as a ratio of two integers having 76 as the numerator and 5 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.