What is 84.12999... as a Fraction?

84.129 equals 8413100 as a fraction.

How do you turn 84.129 repeating into a fraction?

Detailed Answer:

Step 1: To convert 84.129 repeating into a fraction, begin writing this simple equation:

n = 84.129 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 84129.9 (equation 2)

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

1000 × n = 84129.9
 100 × n = 84.129
 900 × n = 75717

75717900 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 9 (the GCF - greatest common factor).

n = 75717900 = 75717 ÷ 9900 ÷ 9 = 8413100. So,

84.129 = 8413100 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 8413100 is also equal to 8413100 when expressed as a mixed number.

The repeating decimal 84.129 (vinculum notation) has a repeated block length of 1. It is also represented as 84.12999... (ellipsis notation) which equals approximately 84.1299999 (decimal approximation)(*).

The recurring decimal 84.129 can be written as a ratio of two integers having 8413 as the numerator and 100 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.