What is 84.14999... as a Fraction?

84.149 equals 168320 as a fraction.

How do you turn 84.149 repeating into a fraction?

Detailed Answer:

Step 1: To convert 84.149 repeating into a fraction, begin writing this simple equation:

n = 84.149 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (14). Multiply both sides by 10³ = 1000.

1000 × n = 84149.9 (equation 2)

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

1000 × n = 84149.9
 100 × n = 84.149
 900 × n = 75735

75735900 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 45 (the GCF - greatest common factor).

n = 75735900 = 75735 ÷ 45900 ÷ 45 = 168320. So,

84.149 = 168320 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 168320 is also equal to 84320 when expressed as a mixed number.

The repeating decimal 84.149 (vinculum notation) has a repeated block length of 1. It is also represented as 84.14999... (ellipsis notation) which equals approximately 84.1499999 (decimal approximation)(*).

The recurring decimal 84.149 can be written as a ratio of two integers having 1683 as the numerator and 20 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.