What is 40.10555... as a Fraction?

40.105 equals 7219180 as a fraction.

How do you turn 40.105 repeating into a fraction?

Detailed Answer:

Step 1: To convert 40.105 repeating into a fraction, begin writing this simple equation:

n = 40.105 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (5) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 40105.5 (equation 2)

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

1000 × n = 40105.5
 100 × n = 40.105
 900 × n = 36095

36095900 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 5 (the GCF - greatest common factor).

n = 36095900 = 36095 ÷ 5900 ÷ 5 = 7219180. So,

40.105 = 7219180 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 7219180 is also equal to 4019180 when expressed as a mixed number.

The repeating decimal 40.105 (vinculum notation) has a repeated block length of 1. It is also represented as 40.10555... (ellipsis notation) which equals approximately 40.1055555 (decimal approximation)(*).

The recurring decimal 40.105 can be written as a ratio of two integers having 7219 as the numerator and 180 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.