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Question 1132164: Express 0.0038(8 repeating) as a fraction in lowest terms and give the sum of the numerator and denominator.
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the fraction in decimal form is .003888888888...... (the 8 repeats endlessly).
let n = the number.
therefore, n= .03888888888........
multiply n by 10,000 to get 10,000 * n = 38.888888888.....
multiply n by 1,000 to get 1,000 * n = 3.88888888......
subtract 1,000 * n from 10,000 * n to get 9,000 * n = 35
divide both sides of this equation by 9,000 to get n = 35 / 9,000
divide numerator and denominator by 5 to get n = 7 / 1800.
that's your fraction in simplified form.
the sum of the simplified numerator and the denominator is 1807.
you can confirm by dividing 7 by 1800 in your calculator.
the answer will be .00388888888.....
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Let me expand on the response provided by tutor @Theo and also offer a shortcut method.
The given number is n = 0.003888888...
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....
Then, as shown in @Theo's response, subtracting the two equations gives 9000n = 35, leading to the fraction 35/9000 = 7/1800.
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.
The (possibly unsimplified) fraction form of the given repeating decimal is found as follows:
(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.
If you want to learn the shortcut, here is another example, using both the formal algebraic method and the shortcut.
n = 0.01234343434... (2 repeating digits "34"; 3 non-repeating digits "012"
Formal algebra....
1000n = 12.34343434...; 100000n = 1234.34343434...
This leads to
99000n = 1234-12 = 1222 --> n = 1222/99000
Shortcut....
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)
The fraction is 1222/99000
Of course, with either method the fraction should be simplified to lowest terms for a final answer.
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