What is 84.14222... as a Fraction?

84.142 equals 18932225 as a fraction.

How do you turn 84.142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 84.142 repeating into a fraction, begin writing this simple equation:

n = 84.142 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 2 digits in the non-repeating part (14). Multiply both sides by 10³ = 1000.

1000 × n = 84142.2 (equation 2)

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

1000 × n = 84142.2
 100 × n = 84.142
 900 × n = 75728

75728900 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 = 75728900 = 75728 ÷ 4900 ÷ 4 = 18932225. So,

84.142 = 18932225 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 18932225 is also equal to 8432225 when expressed as a mixed number.

The repeating decimal 84.142 (vinculum notation) has a repeated block length of 1. It is also represented as 84.14222... (ellipsis notation) which equals approximately 84.1422222 (decimal approximation)(*).

The recurring decimal 84.142 can be written as a ratio of two integers having 18932 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.