What is 165.21777... as a Fraction?

165.217 equals 37174225 as a fraction.

How do you turn 165.217 repeating into a fraction?

Detailed Answer:

Step 1: To convert 165.217 repeating into a fraction, begin writing this simple equation:

n = 165.217 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (21). Multiply both sides by 10³ = 1000.

1000 × n = 165217.7 (equation 2)

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

1000 × n = 165217.7
 100 × n = 165.217
 900 × n = 148696

148696900 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 = 148696900 = 148696 ÷ 4900 ÷ 4 = 37174225. So,

165.217 = 37174225 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 37174225 is also equal to 16549225 when expressed as a mixed number.

The repeating decimal 165.217 (vinculum notation) has a repeated block length of 1. It is also represented as 165.21777... (ellipsis notation) which equals approximately 165.2177777 (decimal approximation)(*).

The recurring decimal 165.217 can be written as a ratio of two integers having 37174 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.