What is 40.11777... as a Fraction?

40.117 equals 18053450 as a fraction.

How do you turn 40.117 repeating into a fraction?

Detailed Answer:

Step 1: To convert 40.117 repeating into a fraction, begin writing this simple equation:

n = 40.117 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (11). Multiply both sides by 10³ = 1000.

1000 × n = 40117.7 (equation 2)

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

1000 × n = 40117.7
 100 × n = 40.117
 900 × n = 36106

36106900 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 = 36106900 = 36106 ÷ 2900 ÷ 2 = 18053450. So,

40.117 = 18053450 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 18053450 is also equal to 4053450 when expressed as a mixed number.

The repeating decimal 40.117 (vinculum notation) has a repeated block length of 1. It is also represented as 40.11777... (ellipsis notation) which equals approximately 40.1177777 (decimal approximation)(*).

The recurring decimal 40.117 can be written as a ratio of two integers having 18053 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.