Write the following as a fraction and show your work. 0.83 (the three is repeating) I don't know how to do it with only one of the numbers repeating and it is not in my book.
Here is a neat algorithm to change any repeating decimal into an exact fraction
Assume you have only the decimal portion of your number, ie, for 4.5676767... consider only the 5676767..
For the numerator, write down all the digits to the end of your repeat, subtract from that the part that does not repeat.
For the denominator, write down a 9 for each digit in the repeating loop, followed by the number of digits that don't repeat.
e.g. 0.45676767..
=(4567-45)/9900
=4522/9900, now reduce to
2261/9900
your number would be
.833333. = (83-8)/90
= 75/90
= 5/6
To write the number 0.83 with the 3 repeating as a fraction, you can use the method of infinite geometric series.
Let's denote the repeating number 3 as x.
Step 1: Recognize the pattern.
If we look closely at the number 0.83, we can observe that the digit 3 repeats indefinitely after the decimal point. So, we can write this number as the sum of two parts: the non-repeating part (0.8) and the repeating part (0.03).
0.83 = 0.8 + 0.03
Step 2: Convert the repeating part to a fraction.
Since the repeating part is 0.03, we can express it as a fraction by dividing it by a number consisting of as many 9's as there are repeating digits. In this case, there is one repeating digit, so our divisor will be 9.
0.03 / 9 = 0.003
Step 3: Combine the non-repeating part and the repeating part.
To combine the non-repeating part (0.8) and the repeating part (0.003), we add them together:
0.8 + 0.003 = 0.803
Step 4: Express the combined number as a fraction.
Now, to write 0.803 as a fraction, we have to count the number of decimal places after the decimal point. In this case, there are 3 decimal places.
So, we multiply 0.803 by 1000 (10 raised to the power of 3) to shift the decimal point:
0.803 × 1000 = 803
Therefore, the fraction representation of 0.83 (with the 3 repeating) is:
0.83 = 803/1000