Can someone explain how you could convert a repeating, nonterminating decimal to a fraction?
Thanks. :)
I will illustrate with an example
4.5676767...
let's just work on the decimal .5676767...
let x = .5676767...
multiply by 100 , (because 2 digits repeat, so 2 zeros in 100)
100x = 56.7676767
subtract ...
99x = 56.2
x = 56.2/99 = 562/990 = 281/495
so 4.5676767.. = 4 281/495 or 2261/495
quick way:
for the numerator,
--->write down all the digits to the end of the first repeat ---- 567, subtract the non-repeating digits
567-5 = 562
for the denominator, for one complete period, we have 1 leading non-repeating followed by 2 repeating ,so
put down a 9 for each repeating digit (so 2 9's) followed by a 0 for each non-repeating digit, (so one 0)
or 990
.567676767... = 562/990 = 281/495
e.g. .34123123123...
- (34123 - 34)/99900 = 34089/99900
how are you people
Sure! To convert a repeating, nonterminating decimal to a fraction, you can use a method called "converting repeating decimals to fractions".
Here is a step-by-step process to convert a repeating decimal to a fraction:
Step 1: Identify the repeating part of the decimal. It can be a single digit, a group of digits, or even just one digit repeated infinitely.
Step 2: Assign a variable to the repeating part. Let's call it "x".
Step 3: Determine the number of repeating digits in your decimal. If you have more than one repeating digit, put them together as the repeating part "x".
Step 4: Construct an equation to represent the decimal in terms of "x". For example, if your decimal is 0.333..., you can write it as "0.x = x". If your decimal is 0.1212..., you can write it as "0.x = 12x".
Step 5: Solve the equation for "x". This is done by multiplying both sides of the equation by a power of 10 to eliminate the decimal places. For example, if you have one repeating digit, multiply both sides by 10. If you have two repeating digits, multiply both sides by 100, and so on.
Step 6: Simplify the equation and solve for "x". Subtract the equation obtained in Step 5 from the original equation obtained in Step 4. Simplify the resulting equation to solve for "x".
Step 7: Express the repeating decimal as a fraction. To do this, place the value of "x" obtained in Step 6 as the numerator of the fraction. The denominator will depend on the number of repeating digits. If you have one repeating digit, use 9 as the denominator. If you have two repeating digits, use 99. If you have three repeating digits, use 999, and so on.
Step 8: Simplify the fraction, if possible, by canceling out any common factors between the numerator and denominator.
Step 9: Write the fraction in its simplest form.
By following these steps, you can convert a repeating, nonterminating decimal into a fraction.