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 N be the number, and let n be the number of digits in the repeating block. For example,
has 2 digits in the repeating block, whereas
has only 1.
Setting N equal to its value, multiply both sides of the equation by 10n. Then subtract N from both sides. You will end up with an equation which you can solve for N 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 102 = 100.
Now subtract N from both sides. Notice that the repeating part cancels out.
Divide both sides by 99, multiply numerator and denominator by a power of 10 to get rid of the decimal point, and simplify.