What is 102.1777... as a Fraction?

102.17 equals 459845 as a fraction.

How do you turn 102.17 repeating into a fraction?

Detailed Answer:

Step 1: To convert 102.17 repeating into a fraction, begin writing this simple equation:

n = 102.17 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 1 digit in the non-repeating part (1). Multiply both sides by 10² = 100.

100 × n = 10217.7 (equation 2)

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

100 × n = 10217.7
 10 × n = 102.17
 90 × n = 9196

919690 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 = 919690 = 9196 ÷ 290 ÷ 2 = 459845. So,

102.17 = 459845 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 459845 is also equal to 102845 when expressed as a mixed number.

The repeating decimal 102.17 (vinculum notation) has a repeated block length of 1. It is also represented as 102.1777... (ellipsis notation) which equals approximately 102.177777 (decimal approximation)(*).

The recurring decimal 102.17 can be written as a ratio of two integers having 4598 as the numerator and 45 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.