What is 89.10888... as a Fraction?

89.108 equals 40099450 as a fraction.

How do you turn 89.108 repeating into a fraction?

Detailed Answer:

Step 1: To convert 89.108 repeating into a fraction, begin writing this simple equation:

n = 89.108 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (8) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 89108.8 (equation 2)

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

1000 × n = 89108.8
 100 × n = 89.108
 900 × n = 80198

80198900 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 2 (the GCF - greatest common factor).

n = 80198900 = 80198 ÷ 2900 ÷ 2 = 40099450. So,

89.108 = 40099450 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 40099450 is also equal to 8949450 when expressed as a mixed number.

The repeating decimal 89.108 (vinculum notation) has a repeated block length of 1. It is also represented as 89.10888... (ellipsis notation) which equals approximately 89.1088888 (decimal approximation)(*).

The recurring decimal 89.108 can be written as a ratio of two integers having 40099 as the numerator and 450 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.