Deriving Fractions from DecimalsDate: 07/19/2001 at 00:03:33 From: Amanda Subject: Percents and Fractions Dear Dr. Math, I am trying to re-learn math. I am currently reviewing percents and fractions. Here is my question: how do I get the fraction in problems like: 12.5% = 0.125--????????? = 1/8 66.7% = 0.667--?????????? = 2/3 I don't understand how they come up with this. Your help would be greatly appreciated. Thanks! Amanda Date: 07/19/2001 at 16:18:38 From: Doctor Ian Subject: Re: Percents and Fractions Hi Amanda, It's easy to forget that a decimal is just a shorthand way of writing a fraction. For example, 0.125 = 125 / 1000 The two expressions mean exactly the same thing. Now, if you start with the fraction 125/1000 and try to simplify it, 125 5 * 5 * 5 ---- = --------------------- 1000 2 * 2 * 2 * 5 * 5 * 5 the 5's in the numerator and denominator cancel out, leaving you with 125/1000 = 1/8. In the same way, you can convert something like 0.5625 = 5625/10000 3 * 3 * 5 * 5 * 5 * 5 = ----------------------------- 2 * 2 * 2 * 2 * 5 * 5 * 5 * 5 3 * 3 = ------------- 2 * 2 * 2 * 2 = 9/16 In the case of a repeating decimal, you use (10...0 - 1) in the denominator, rather than 10...0. Here's why: x = 0.666... 10x = 6.666... 10x - x = 6 9x = 6 x = 6/9 So the terminating decimal 0.667 is the same as 667/1000, but the repeating decimal 0.666... is the same as 6/9, or 66/99, or 666/999. And all of those are equal to 2/3. Here's another example: x = 0.521521... 1000x = 521.521521... 1000x - x = 521 999x = 521 x = 521/999 Note that 66.7% = 0.667 = 2/3 (False!) is NOT a true equation! It's only true if the percentage is a repeating decimal, i.e., 66.6...% = 0.666... = 2/3 (True!) Having said all of this, a handful of fractions appear so often that it will simplify your life considerably to memorize their decimal equivalents: 1/2 = 0.5 1/3 = 0.333... 1/4 = 0.25 1/5 = 0.2 1/8 = 0.125 One you know these fractions, you can compute others (like 3/8) by multiplying by the appropriate numerator, e.g., 3/8 = 3 * (1/8) = 3 * (0.125) = 0.375 I hope this helps. Let me know if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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