What is 77.13888... as a Fraction?

77.138 equals 277736 as a fraction.

How do you turn 77.138 repeating into a fraction?

Detailed Answer:

Step 1: To convert 77.138 repeating into a fraction, begin writing this simple equation:

n = 77.138 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (8) and 2 digits in the non-repeating part (13). Multiply both sides by 10³ = 1000.

1000 × n = 77138.8 (equation 2)

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

1000 × n = 77138.8
 100 × n = 77.138
 900 × n = 69425

69425900 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 25 (the GCF - greatest common factor).

n = 69425900 = 69425 ÷ 25900 ÷ 25 = 277736. So,

77.138 = 277736 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 277736 is also equal to 77536 when expressed as a mixed number.

The repeating decimal 77.138 (vinculum notation) has a repeated block length of 1. It is also represented as 77.13888... (ellipsis notation) which equals approximately 77.1388888 (decimal approximation)(*).

The recurring decimal 77.138 can be written as a ratio of two integers having 2777 as the numerator and 36 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.