What is 89.21333... as a Fraction?

89.213 equals 669175 as a fraction.

How do you turn 89.213 repeating into a fraction?

Detailed Answer:

Step 1: To convert 89.213 repeating into a fraction, begin writing this simple equation:

n = 89.213 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (21). Multiply both sides by 10³ = 1000.

1000 × n = 89213.3 (equation 2)

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

1000 × n = 89213.3
 100 × n = 89.213
 900 × n = 80292

80292900 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 12 (the GCF - greatest common factor).

n = 80292900 = 80292 ÷ 12900 ÷ 12 = 669175. So,

89.213 = 669175 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 669175 is also equal to 891675 when expressed as a mixed number.

The repeating decimal 89.213 (vinculum notation) has a repeated block length of 1. It is also represented as 89.21333... (ellipsis notation) which equals approximately 89.2133333 (decimal approximation)(*).

The recurring decimal 89.213 can be written as a ratio of two integers having 6691 as the numerator and 75 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.