What is 40.11666... as a Fraction?

40.116 equals 240760 as a fraction.

How do you turn 40.116 repeating into a fraction?

Detailed Answer:

Step 1: To convert 40.116 repeating into a fraction, begin writing this simple equation:

n = 40.116 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (11). Multiply both sides by 10³ = 1000.

1000 × n = 40116.6 (equation 2)

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

1000 × n = 40116.6
 100 × n = 40.116
 900 × n = 36105

36105900 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 15 (the GCF - greatest common factor).

n = 36105900 = 36105 ÷ 15900 ÷ 15 = 240760. So,

40.116 = 240760 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 240760 is also equal to 40760 when expressed as a mixed number.

The repeating decimal 40.116 (vinculum notation) has a repeated block length of 1. It is also represented as 40.11666... (ellipsis notation) which equals approximately 40.1166666 (decimal approximation)(*).

The recurring decimal 40.116 can be written as a ratio of two integers having 2407 as the numerator and 60 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.