Multiplying Mixed NumbersDate: 04/06/2004 at 12:55:59 From: Jane Subject: Multiplying mixed numbers Why do you need to convert mixed numbers to improper fractions before multiplying them together? I teach 5th grade mathematics, and the children asked me about this. We understand that you have to do it, but what is the mathmatical theory behind this operation? I have looked in numerous math resources and cannot find an explanation. Date: 04/06/2004 at 13:50:53 From: Doctor Peterson Subject: Re: Multiplying mixed numbers Hi, Jane. This is an excellent question! Like many things in math, converting to improper fractions isn't really NECESSARY, just CONVENIENT (though you might not think so until you finish reading this!). You can multiply mixed numbers directly if you want, just the same way you multiply whole numbers. The key is the distributive property: to multiply a sum, you can multiply each part of the sum and then add. With whole numbers, this looks like 25 * 32 = (20 + 5)*32 = 20*32 + 5*32 = 20*(30 + 2) + 5*(30 + 2) = 20*30 + 20*2 + 5*30 + 5*2 = 600 + 40 + 150 + 10 = 800 This is really what we are doing when we write 32 * 25 ---- 160 <- 5*32 64 <- 20*32 ---- 800 With mixed numbers, you can do the same thing if you want: (2 1/2)*(1 1/5) = (2 + 1/2)*(1 + 1/5) = 2*(1 + 1/5) + 1/2*(1 + 1/5) = 2*1 + 2*1/5 + 1/2*1 + 1/2*1/5 = 2 + 2/5 + 1/2 + 1/10 = 2 + 4/10 + 5/10 + 1/10 = 2 + 10/10 = 3 You could write this something like this: 2 1/2 * 1 1/5 -------- 1/10 = 1/10 2/5 = 4/10 2 1/2 = 5/10 --------------- 2 10/10 = 3 Or, you can convert to improper fractions: 5/2 * 6/5 = (5*6)/(2*5) = 6/2 = 3 Now why is this so much easier? It's because when you mix multiplication and addition, you have to use the distributive property, which gives you a lot of parts to add up. When you mix multiplication with pure fractions (no addition), everything works out neatly because fractions are the same thing as division, and multiplication and division are just different sides of the same operation. So mixed numbers, which are sums, work well when you want to add or subtract; while pure fractions, which are divisions, work well when you want to multiply or divide. We convert to whatever form works best for what we want to do. 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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