What is 84.16666... as a Fraction?

84.166 equals 5056 as a fraction.

How do you turn 84.166 repeating into a fraction?

Detailed Answer:

Step 1: To convert 84.166 repeating into a fraction, begin writing this simple equation:

n = 84.166 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (16). Multiply both sides by 10³ = 1000.

1000 × n = 84166.6 (equation 2)

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

1000 × n = 84166.6
 100 × n = 84.166
 900 × n = 75750

75750900 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 150 (the GCF - greatest common factor).

n = 75750900 = 75750 ÷ 150900 ÷ 150 = 5056. So,

84.166 = 5056 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 5056 is also equal to 8416 when expressed as a mixed number.

The repeating decimal 84.166 (vinculum notation) has a repeated block length of 1. It is also represented as 84.16666... (ellipsis notation) which equals approximately 84.1666666 (decimal approximation)(*).

The recurring decimal 84.166 can be written as a ratio of two integers having 505 as the numerator and 6 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.