Dividing Decimals with a RemainderDate: 03/30/2001 at 16:33:56 From: Genna Subject: Dividing Decimals Dear Dr. Math, I am a student in an elementary school and just got accepted to be in accelerated math next year in Jr. High. I have a question about dividing decimals. I know how to divide decimals, but when there is a remainder my teacher tells me I have to add zeros to the inside number. He says you can either add 2 or 3. I usually add 2 zeros, but if you add 2 you will get a totally different answer than if you add 3 zeros. Also, sometimes when you have a remainder and you bring it out 2 or 3 places (or however many it tells you to), you still have a remainder. My teacher says to just leave the remainder and round the answer, but I don't understand. If you could explain to me what to do when the directions only say DIVIDE: and you get a remainder, and also if the directions say ROUND TO THE HUNDRETHS PLACE: what to do when there is still a remainder, I would be very happy. Thank you very much, Genna Date: 03/30/2001 at 23:29:15 From: Doctor Peterson Subject: Re: Dividing Decimals Hi, Genna. Let's take an example, so we have something specific to talk about: ___1.9_ 23 ) 45.6 23 -- 22 6 20 7 ---- 1 9 Okay, we have a remainder, and we want to round our answer to the hundredths place, so we'll add two zeros to the dividend and continue. (I'll deal with how many zeros to add in a moment.) ___1.982_ 23 ) 45.600 23 -- 22 6 20 7 ---- 1 90 1 84 ---- 60 46 -- 14 Now what do we do? Since we were told to round to two decimal places, we just look at the 2 in the quotient, drop it, and leave the rest as it is. The answer is 1.98. I didn't really have to add that last zero. I could have stopped with the remainder 6, and seen that the next digit of the quotient would be less than 5, because 6 is less than half of 23. See if you can see why that is true. (I don't know what you mean when you say that you get a totally different answer if you add two zeros or three. I just see one extra decimal place in the answer, which is not very different. Can you give me an example of what you mean?) Now let's get back to the big question: what is this all about? Why should you just ignore the remainder? The problem with decimals is that most division problems never end. You can keep adding more zeros to the dividend, and you'll just get more zeros in the quotient. If you've learned about repeating decimals, you know why that is: when you change a fraction to a decimal, you get a terminating decimal only if the denominator (in lowest terms) has only 2 and 5 as prime factors, so that it can be converted to a fraction with a power of ten in the denominator. So when we work with decimals, we expect to have to approximate. The reason we can approximate without losing anything important is that each decimal place we get is worth a tenth of the previous one, so eventually they get small enough not to affect whatever we are going to do with them. At that point we can just drop the remainder (and therefore all the rest of the digits), and round if we wish. An important topic related to this is "significant digits." You may want to search our archives for this phrase and read about it; it explains how we can decide how many digits we need, and why digits after a certain point don't matter and can be ignored. When we need to be precise, we use fractions rather than decimals, because we never have to drop anything. When we use decimals, we KNOW that we are going to be rounding, so it doesn't bother us. Write back if you have any further questions. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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