How do you turn a decimal into a fraction or mixed fraction?

Answer 1

A mixed number will occur if there is a number other than 0 before the decimal point. When this occurs, this number (before the decimal point) is the whole number of a mixed fraction.

Having dealt with the whole number (if any), the remaining part after the decimal point needs to be changed into a fraction. There are four type of decimal number that occur:

• If the decimal is a terminating decimal, that is it ends and does not go on forever:

Put the digits after the decimal point over 1 followed by the same number of zeros and reduce to simplest form.

examples:

• 0.125

= 125/1000 (3 digits in '125', so three 0s after the 1)

= 1/8 (divide top and bottom by 125)

• 4.35

= 435/100 ('4' whole number, 2 digits in '35' so two 0s after the 1)

= 47/20 (divide top and bottom of fraction by 5)

• If the decimal is a recurring decimal that repeats a number of digits immediately after the decimal point:

Put the digits that repeat over the same number of 9s as there are digits repeating and reduce to simplest form.

examples

• 0.666.... (the digit '6' repeating forever)

= 6/9 (1 digit repeats, thus one 9)

= 2/3 (divide top and bottom by 3)

• 0.3636... (the digits '36' repeating forever)

= 36/99 (2 digits repeat, thus two 9s)

= 4/11 (divide top and bottom by 9)

• 0.57142857... (the digits '571428' repeating forever)

= 571428/999999 (6 digits repeat, thus six 9s)

= 4/7 (divide top and bottom by 142857)

• If the decimal is a recurring decimal that repeats a number of digits, but there are also some digits immediately after the decimal point that do not repeat:

A combination of the first two methods - put the non-recurring digits over 1 followed by the same number of zeros; put the recurring digits over the same number of nines, followed by the same number of zeros as the first fraction had. Then add the two fractions together and reduce to simplest terms:

examples

• 0.8333... (the '3' repeats forever)

= 8/10 + 3/90 (one digit in '8', so one 0 after 1, one digit (3) repeats so one 9 but also one 0 from first fraction)

= 8/10 + 1/30 (reducing the second fraction - divide top and bottom by 3)

= (24 + 1)/30 (adding the fractions, lcd = 30)

= 25/30

= 5/6 (divide top and bottom by 5)

• 0.28333... (the '3' repeats forever)

= 28/100 + 3/900 (two digits in '25', so two 0s after 1, one digit (3) repeats so one 9 but also two 0s from first fraction)

= 28/100 + 1/300 (reducing the second fraction - divide top and bottom by 3)

= (84 + 1)/300 (adding the fractions, lcd = 300)

= 85/300

= 17/60 (divide top and bottom by 5)

• 0.15513513... (the '513' repeats forever)

= 15/100 + 513/99900 (two digits in '15', so two 0s after 1, three digit (513) repeat so three 9s but also two 0s from first fraction)

= 15/100 + 19/3700 (reducing the second fraction - divide top and bottom by 27)

= (555 + 19)/3700 (adding the fractions, lcd = 3700)

= 574/3700

= 287/1850 (divide top and bottom by 2)

• If the decimal does not end and does not have any recurring digits

This type of decimal is an irrational decimal and cannot be represented as a fraction.

examples

• 3.1415926... (= π)
• 1.4142135... (= √2)