How do you change a decimal to a mixed number or fractions?

Answer 1

If the decimal goes on forever without repeating, the decimal is the representation of an irrational number and cannot be expressed as a fraction. Otherwise:

Any number (other then zero) before the decimal point will be the whole number of a mixed number, and the digits after the decimal point represent the fraction.

If the decimal terminates, then take the digits after the decimal point and put them over a '1' followed by the same number of '0's as there are digits, and simplify.

eg 0.125 has three digits after the decimal point, so put the three digits (123) over a '1' followed by three '0's. that is over '1000' and simplify:

0.125 = 125/1000 = 25/200 = 5/40 = 1/8

If the decimal does not terminate but repeats a sequence of digits, put the repeating digits over the same number of '9's and simplify.

eg 0.121212... has two repeating digits (12), so put them over two '9's (99) and simplify:

0.121212... = 12/99 = 4/33

If the decimal starts with a few digits and then repeats, convert the first few digits to a fraction as above (for the terminating decimal) and add the repeating digits converted to a fraction as above, but also follow the '9's of the repeating fraction by the same number of '0's as the initial digits.

eg 0.1666... starts with one digit (1) followed by one repeating digit (6):

The one non-repeating digit becomes 1/10 (the denominator is '1' followed by one '0').

The one repeating digit becomes 6/90 (the denominator is '9' as there is one repeating digit, followed by one '0' as there was one non-repeating digit). Thus:

0.1666... = 1/10 + 6/90

= 9/90 + 6/90

= 15/90 = 5/30 = 1/6

Another example of this "mixed" non-recurring and recurring decimal:

0.41666...

Two non-repeating digits (41) → 41/100

One repeating digit (6) → 6/900 (the one '9' as one repeating digit, two '0's as two non-repeating digits)

→ 0.41666... = 41/100 + 6/900

= 123/300 + 2/300

= 125/300 = 25/60 = 5/12

Another example of this "mixed" non-recurring and recurring decimal:

0.4181818...

One non-repeating digit (4) → 4/10

Two repeating digits (18) → 18/990 (the two '9' as two repeating digits, one '0' as one non-repeating digit)

→ 0.4181818... = 4/10 + 18/990

= 44/110 + 2/110

= 46/110 = 23/55