What is 101.5111... as a Fraction?

101.51 equals 456845 as a fraction.

How do you turn 101.51 repeating into a fraction?

Detailed Answer:

Step 1: To convert 101.51 repeating into a fraction, begin writing this simple equation:

n = 101.51 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (1) and 1 digit in the non-repeating part (5). Multiply both sides by 10² = 100.

100 × n = 10151.1 (equation 2)

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

100 × n = 10151.1
 10 × n = 101.51
 90 × n = 9136

913690 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 = 913690 = 9136 ÷ 290 ÷ 2 = 456845. So,

101.51 = 456845 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 456845 is also equal to 1012345 when expressed as a mixed number.

The repeating decimal 101.51 (vinculum notation) has a repeated block length of 1. It is also represented as 101.5111... (ellipsis notation) which equals approximately 101.511111 (decimal approximation)(*).

The recurring decimal 101.51 can be written as a ratio of two integers having 4568 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.