Changing Decimals to Fractions and ReducingDate: 12/12/2002 at 20:42:29 From: Bihanca Subject: Changing decimals to fractions Hi, Could you please help me to understand, in the simplest way, how to write a decimal as a lowest-term fraction or mixed number? Example: 4.35 I'm stuck after this. I just don't get it. Thank you! Date: 12/12/2002 at 22:13:38 From: Doctor Ian Subject: Re: Changing decimals to fractions Hi Bihanca, The absolute simplest way is this. First, separate the integer and decimal parts, like so: 375 4.375 ---> 4 --- Now, for each digit in the decimal part, put a (2*5) in the denominator: 375 4.35 ---> 4 ----------------- (2*5)(2*5)(2*5) Now start trying to divide those 2's and 5's into the numerator, to cancel them out. For example, 375 is divisible by 5, so go ahead and do the division: 75 4 -------------- (2*5)(2*5)(2) 75 is also divisible by 5: 15 4 -------------- (2*5)(2)(2) And 15 is divisible by 5: 3 4 ----------- (2)(2)(2) There are no more divisions to do, so you're done: 3 4 - 8 Does it look kind of like a magic trick? If so, think about what we did with the fractional part. When we write something like 0.375 this is really just a very compact way of writing something much bigger: 3 7 5 0.375 is the same as -- + --- + ---- 10 100 1000 (If you didn't already know this, make sure you understand it now. And if you're having trouble making sense of it, please let me know, because it's a really crucial thing to understand.) Now, note that 3 7 5 300 70 5 -- + --- + ---- = ---- + ---- + ---- 10 100 1000 1000 1000 1000 300 + 70 + 5 = -------------- 1000 375 = ------ 1000 375 = ---------- 10*10*10 375 = ---------------- (2*5)(2*5)(2*5) So all I did was skip a few steps and go right to the prime factorization of the denominator, which makes it easier to do the reduction. If you don't have to worry about reducing the fraction to lowest terms, it's even easier. Just separate the integer and decimal parts, 375 4 ----- and write a zero for each digit in the denominator: 375 4 ----- 000 Then put a '1' in front: 375 4 ---- 1000 There is one final thing to watch out for. If you have any leading zeros, you have to include them as digits during the first step: 035 4.035 ---> 4 --------------- (2*5)(2*5)(2*5) Otherwise you'd end up treating 4.35 and 4.035 as if they were the same number, which they're not. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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