What is 15.80222... as a Fraction?

15.802 equals 7111450 as a fraction.

How do you turn 15.802 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.802 repeating into a fraction, begin writing this simple equation:

n = 15.802 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 2 digits in the non-repeating part (80). Multiply both sides by 10³ = 1000.

1000 × n = 15802.2 (equation 2)

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

1000 × n = 15802.2
 100 × n = 15.802
 900 × n = 14222

14222900 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 = 14222900 = 14222 ÷ 2900 ÷ 2 = 7111450. So,

15.802 = 7111450 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 7111450 is also equal to 15361450 when expressed as a mixed number.

The repeating decimal 15.802 (vinculum notation) has a repeated block length of 1. It is also represented as 15.80222... (ellipsis notation) which equals approximately 15.8022222 (decimal approximation)(*).

The recurring decimal 15.802 can be written as a ratio of two integers having 7111 as the numerator and 450 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.