How do you convert decimals to a fraction?

Answer 1

There are three different situations, corresponding to the three types of decimal numbers: terminating, repeating and those which are neither terminating nor repeating.

Terminating: If the decimal number has d digits after the decimal point, then rename it as a fraction whose numerator is the decimal number without the decimal point, and the denominator is 10d or 1 followed by d zeros.

For example, 34.567

d = 3 so the denominator is 1000.

and the fraction is 34567/1000.

Repeating: Until you become expert at this I suggest you do this in two stages (using c and d separately). Suppose there are c digits after the decimal place where the digits are non-repeating, after which you get a repeating pattern of a string of d digits. Then the numerator is the old original string including one lot of the repeated digits minus the original string with none of the repeating digits. The denominator is 10c*(10d - 1), which is a string of d 9s followed by c 0s.

For example

123.26159159… There are 2 digits, "26", after the decimal point before the repeats kick in so c = 2, and the repeating string "159" is 3 digits long so d = 3.

So the numerator is 12326159 – 12326 = 12313833

and the denominator is 99900

Therefore the fraction is 12313833/99900.

Non-terminating and non-repeating: There is no way to get a proper fraction since, by definition, this is an irrational number. The best that you can do is to round it to a suitable number of digits and then treat that answer as a terminating decimal.

In all cases, you should check to see if the fraction can be simplified.