What is 78.12888... as a Fraction?

78.128 equals 17579225 as a fraction.

How do you turn 78.128 repeating into a fraction?

Detailed Answer:

Step 1: To convert 78.128 repeating into a fraction, begin writing this simple equation:

n = 78.128 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (8) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 78128.8 (equation 2)

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

1000 × n = 78128.8
 100 × n = 78.128
 900 × n = 70316

70316900 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 4 (the GCF - greatest common factor).

n = 70316900 = 70316 ÷ 4900 ÷ 4 = 17579225. So,

78.128 = 17579225 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 17579225 is also equal to 7829225 when expressed as a mixed number.

The repeating decimal 78.128 (vinculum notation) has a repeated block length of 1. It is also represented as 78.12888... (ellipsis notation) which equals approximately 78.1288888 (decimal approximation)(*).

The recurring decimal 78.128 can be written as a ratio of two integers having 17579 as the numerator and 225 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.