What is 1.65666... as a Fraction?

1.656 equals 497300 as a fraction.

How do you turn 1.656 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.656 repeating into a fraction, begin writing this simple equation:

n = 1.656 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (65). Multiply both sides by 10³ = 1000.

1000 × n = 1656.6 (equation 2)

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

1000 × n = 1656.6
 100 × n = 1.656
 900 × n = 1491

1491900 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 3 (the GCF - greatest common factor).

n = 1491900 = 1491 ÷ 3900 ÷ 3 = 497300. So,

1.656 = 497300 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 497300 is also equal to 1197300 when expressed as a mixed number.

The repeating decimal 1.656 (vinculum notation) has a repeated block length of 1. It is also represented as 1.65666... (ellipsis notation) which equals approximately 1.6566666 (decimal approximation)(*).

The recurring decimal 1.656 can be written as a ratio of two integers having 497 as the numerator and 300 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.