Determining Which Number to Divide by Which
Date: 10/04/2004 at 08:49:07 From: Dominick Subject: figuring out which is the divisor Julia plans to use a recipe that yields 15 pounds of fudge. She intends to wrap the fudge she makes in 3/4 pound boxes and give one box to each of her friends and relatives for gifts. How many gifts will she have? I don't know how to tell if I calculate 15 divided by 3/4 or if it's 3/4 divided by 15. I get it confused sometimes. 15 divided by 3/4 3/4 divided by 15 15/1 divided by 3/4 3/4 divided by 15/1 15/1 times 4/3 3/4 times 1/15 5/1 times 4/1 1/4 times 1/5 20 answer 1/20 answer
Date: 10/04/2004 at 10:06:07 From: Doctor Peterson Subject: Re: figuring out which is the divisor Hi, Dominick. If you look at what you are doing in your two cases, you will see that the answers are reciprocals of one another. But that's not the main issue here. How can you decide what to divide by what? One way is to think about what division means. You want some number of boxes, 3/4 pound each, which will be 15 pounds all together. So you want to find the number by which you can multiply 3/4, to give 15: x * 3/4 = 15 The opposite of this multiplication is division BY the known factor: x = 15 / 3/4 (I'm using "*" to mean multiplication and "/" to mean division.) Thinking in this way makes it clear what you divide by what: to find one factor, you divide the product by the other factor. Along the same lines, once you get an answer you can check it. What is the total weight of 20 3/4-pound boxes? It is 20 times 3/4, which does give 15. Also, just look at the answers. Does it make sense that 1/20 of a box should weigh 15 pounds? No; 1/20 times 3/4 pound gives 3/80 of a pound! That sort of check should be a regular part of your problem-solving method. Another thing that can help is to change the numbers in the problem to avoid fractions, which often makes the answer more obvious. If you wanted to divide 15 pounds into 3-pound packages, how many will you get? This time you can picture making packages until you reach a total of 15 pounds, and you can see quickly that you can find that answer by dividing 15 by 3. Do the same with the real numbers. Finally, often you can use units to help. You have 15 pounds per batch 3/4 pounds per box X boxes per batch That is, you want to find how many 3/4-pound boxes you can make from a 15-pound batch. If you divided the 3/4 by the 15, and kept the units with the calculation, you would get 3/4 pounds 15 pounds 3/4 pounds 1 batch ---------- / --------- = ---------- * --------- = 1/20 batch/box 1 box 1 batch 1 box 15 pounds So your answer of 1/20 is not the number of boxes in a batch, but the number of batches in a box--that is, each box is 1/20 of the batch. And therefore the whole batch is 20 boxes. If you do the calculation the right way, you'll see the answer come out to 20 boxes/batch. If you don't quite follow what I'm doing here, see this page: Unit Conversions, Unit Cancellation http://mathforum.org/library/drmath/sets/select/dm_unit_convert.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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