Three- and Four-Digit Long DivisionDate: 03/15/99 at 20:20:25 From: Candace VanHulle Subject: Long Division Could you explain how to divide problems such as 67932/918 and 5676308/6532? I just do not understand how to go about finding how many times a number goes into another number when you are using three or four digits. Thanks for your help. Date: 03/19/99 at 16:57:30 From: Doctor Peterson Subject: Re: Long Division I am going to assume that you can handle smaller divisors and know the basic techniques of long division, but need help on how to choose (guess) a digit to try in the quotient. Let us try your first problem, and see how it goes. _______ 918)67932 The first set of digits bigger than 918 is 6793. We need to estimate the quotient 6793/918. To get a rough estimate, you can just drop the last two digits from both numbers: 67/9 = 7. (Do you see why? It is the same as approximating the fraction 6793/918 by 6700/900 and simplifying.) So let us try using 7 in the quotient: 7 * 918 = 6426, so we write ____7__ 918)67932 6426 ---- 367 That looks good: the remainder is positive (always a good sign) but less then 918. So we continue: ____7__ 918)67932 6426 ---- 3672 Again, we can estimate 3672/918 by 36/9 = 4. Since the result is a whole number, I will not be surprised if it is wrong; it will not take much increase in the dividend to push the quotient past 4. But let us try it: 4 * 918 = 3672. Let us write it down: ____74_ 918)67932 6426 ---- 3672 3672 ---- 0 That was too easy; it did not let me demonstrate how to recover from a mistake. Since estimation plays a big role here, you have to be prepared to make a wrong guess. Let us suppose I had somehow guessed 6 for the first digit: ____6__ 918)67932 5508 ---- 1285 When I see that my remainder (1285) is bigger than my divisor (918), I know I have to increase the quotient; I will try 7 and it will be right. Suppose instead that I guessed 8: ____8__ 918)67932 7344 ---- I do not even bother subtracting, because 7344 is too big to subtract from 6793; I stop right here and decrease my quotient to 7. I can also give you some more ideas on how to guess in the first place. As I said, for the first digit we approximated the fraction 6793/918 by 6700/900, and found that 7*900 = 6300. This worked out well, because if we add the dropped 18 back onto the 900, and multiply it by 7, we are only adding 126 to the quotient, which is not too much more than the 93 we dropped from 6793. If the digits you are dropping from the divisor are bigger, you may be better off estimating by rounding up. For example, to estimate 6793/983, I would probably use 6700/1000 = 6 rather than 6700/900 = 7. Am I right? 6 * 983 = 5898, while 7 * 983 = 6881, so 7 would have been barely too large. If the dividend had been smaller, say 6719, I would have definitely been right to round the denominator up. If this is where you have trouble, I would recommend two things. First, practice just estimating single-digit quotients like this to get a feel for when you should round up or down. Second, get used to correcting wrong guesses, because that is an unavoidable part of the process. I hope this will build your confidence on these problems. If you get stuck on some specific problem, feel free to write back and show me your work, so I can help you out. And do not forget to check our archives whenever you have a question; here is the page on division: http://mathforum.org/dr.math/tocs/division.elem.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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