What is 99.9777... as a Fraction?

99.97 equals 449945 as a fraction.

How do you turn 99.97 repeating into a fraction?

Detailed Answer:

Step 1: To convert 99.97 repeating into a fraction, begin writing this simple equation:

n = 99.97 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 1 digit in the non-repeating part (9). Multiply both sides by 10² = 100.

100 × n = 9997.7 (equation 2)

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

100 × n = 9997.7
 10 × n = 99.97
 90 × n = 8998

899890 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 = 899890 = 8998 ÷ 290 ÷ 2 = 449945. So,

99.97 = 449945 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 449945 is also equal to 994445 when expressed as a mixed number.

The repeating decimal 99.97 (vinculum notation) has a repeated block length of 1. It is also represented as 99.9777... (ellipsis notation) which equals approximately 99.977777 (decimal approximation)(*).

The recurring decimal 99.97 can be written as a ratio of two integers having 4499 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.