In Part A of this lesson, we saw how to convert a terminating decimal number to a fraction. If a decimal is repeating (in other words, after a while, some pattern of digits repeats over and over), then we have to use a different strategy.
Let be the number, and let be the number of digits in the repeating block. For example,
has digits in the repeating block, whereas
has only .
Setting equal to its value, multiply both sides of the equation by . Then subtract from both sides. You will end up with an equation which you can solve for as a fractional value.
Convert the decimal
into a fraction.
Start with the equation:
There are two digits in the repeating block, so multiply both sides by .
Now subtract from both sides. Notice that the repeating part cancels out.
Divide both sides by , multiply numerator and denominator by a power of to get rid of the decimal point, and simplify.