What is 77.2333... as a Fraction?

77.23 equals 231730 as a fraction.

How do you turn 77.23 repeating into a fraction?

Detailed Answer:

Step 1: To convert 77.23 repeating into a fraction, begin writing this simple equation:

n = 77.23 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 1 digit in the non-repeating part (2). Multiply both sides by 10² = 100.

100 × n = 7723.3 (equation 2)

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

100 × n = 7723.3
 10 × n = 77.23
 90 × n = 6951

695190 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 3 (the GCF - greatest common factor).

n = 695190 = 6951 ÷ 390 ÷ 3 = 231730. So,

77.23 = 231730 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 231730 is also equal to 77730 when expressed as a mixed number.

The repeating decimal 77.23 (vinculum notation) has a repeated block length of 1. It is also represented as 77.2333... (ellipsis notation) which equals approximately 77.233333 (decimal approximation)(*).

The recurring decimal 77.23 can be written as a ratio of two integers having 2317 as the numerator and 30 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.