Question 1132164
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Let me expand on the response provided by tutor @Theo and also offer a shortcut method.<br>
The given number is n = 0.003888888...<br>
In the algebraic method for finding the equivalent fraction, the objective is to find two numbers in which the decimal parts are the same.  In this example, multiplying the given number by 1000 gives 1000n = 3.8888888.....; multiplying by 10000 gives 10000n = 38.8888888....<br>
Then, as shown in @Theo's response, subtracting the two equations gives 9000n = 35, leading to the fraction 35/9000 = 7/1800.<br>
I work a lot with high school students who compete in math competitions where the speed of solving a problem is important.  Here is a shortcut that does the same thing as the formal algebraic method with less work.<br>
The (possibly unsimplified) fraction form of the given repeating decimal is found as follows:<br>
(1) The numerator is the non-repeating digits followed by one cycle of the repeating digits, minus the non-repeating digits:  In this example, 0038-003 = 35.
(2) The denominator is one 9 for each repeating digit followed by one 0 for each non-repeating digit:  In this example, one 9 followed by three 0s = 9000.
The fraction is 35/9000 = 7/1800.<br>
If you want to learn the shortcut, here is another example, using both the formal algebraic method and the shortcut.<br>
n = 0.01234343434... (2 repeating digits "34"; 3 non-repeating digits "012"<br>
Formal algebra....<br>
1000n = 12.34343434...; 100000n = 1234.34343434...<br>
This leads to 
99000n = 1234-12 = 1222 -->  n = 1222/99000<br>
Shortcut....<br>
numerator is 01234-012 = 01222 = 1222  (non-repeating digits followed by one cycle of the repeating digits, minus non-repeating digits)
denominator is 99000  (2 repeating digits; 3 non-repeating)<br>
The fraction is 1222/99000<br>
Of course, with either method the fraction should be simplified to lowest terms for a final answer.