Making Fractions into PercentagesDate: 04/30/2003 at 22:19:52 From: Bob Subject: Percentages I am really confused about how to make fractions into percentages. For example, 3 over 5 is what percent? Can you show me a simple way? Date: 05/01/2003 at 09:30:02 From: Doctor Ian Subject: Re: Percentages Hi Bob, There are three kinds of cases. (1) If the denominator of the fraction evenly divides 100, as in your example of 3/5, then just convert to the equivalent fraction: 3 20 60 - * -- = --- = 60% 5 20 100 Note that 60/100 _is_ 60%. Those are just two different notations for the same thing. (2) If the denominator evenly divides some higher power of 10, start by doing the same thing, 3 125 375 - * --- = ---- 8 125 1000 and then move the decimal point over in the numerator while lopping zero's off the end of the denominator, until you get down to 100: 3 125 375 37.5 - * --- = ---- = ----- = 37.5% 8 125 1000 100 But don't just learn this as a rule. Make sure you understand _why_ this works. How do you _know_ when the denominator will evenly divide 100, or some other power of 10? The easiest way to figure this out is to break the denominator into prime factors. Consider the following cases: Fraction Denominator Prime factors -------- ----------- ------------- 1/2 2 2 1/5 5 5 1/8 8 2*2*2 1/20 20 2*2*5 1/15 15 3*5 Now, for something to be a power of 10, it has to have _only_ pairs of 2's and 5's as prime factors: 10 = 2*5 100 = (2*5)*(2*5) 1000 = (2*5)*(2*5)*(2*5) So if you have a denominator whose prime factors are only 2's and 5's, you can make an equivalent fraction by supplying the 'missing' 2's and 5's. 1 1 2*5*5 50 - = - * ----- = ----------- 2 2 2*5*5 (2*5)*(2*5) 1 1 2*2*5 20 - = - * ----- = ----------- 5 5 2*2*5 (2*5)*(2*5) 1 1 5*5*5 125 - = ----- * ----- = ----------------- 8 2*2*2 5*5*5 (2*5)*(2*5)*(2*5) Does that make sense? (3) Now, what about a denominator like 15? It has a 3 as one of its prime factors, so there's _nothing_ you can multiply it by that will let you end up with _only_ 2's and 5's as prime factors. In this case, you just have to go ahead and divide: 0.0 6 6 ... __________ 15 ) 1.0 0 0 0 9 0 ----- 1 0 0 9 0 ----- 1 0 0 At some point, you have to truncate to get an approximation, e.g., 1/15 = 0.06667 Once you've done that, you're almost done, because a decimal is really just a way of writing a fraction with a power of 10. 0.07 = 7/100 0.067 = 67/1000 0.0667 = 667/10000 and so on. Let's say we settle on 667/10000. Now we move the decimal place and get rid of zeros, as before: 667 66.7 6.67 ----- = ---- = ------ = 6.67% 10000 1000 100 This third case, where you just go ahead and divide, is the most general, and can _always_ be used, if you don't feel like taking the time to find the prime factors of your denominator. Does this all make sense? Do you think you can work these kinds of problems on your own now? (A good way to find out whether you understand something is to try to explain it to someone who doesn't already know it.) Write back if any of this wasn't clear, or if you have other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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