Multiplying Mixed NumbersDate: 06/02/2003 at 16:07:16 From: Natasha Subject: Mixed numbers Why can you not multiply mixed numbers without changing them into improper fractions? I understand that it is easier and I understand the process, but is there a theory? Date: 06/04/2003 at 14:32:21 From: Doctor Douglas Subject: Re: Mixed numbers Hi Natasha, Thanks for writing to the Math Forum. You can indeed multiply mixed numbers without converting them to improper fractions. 3 1/2 x 4 1/5 = (3 + 1/2) x (4 + 1/5) = (3 x 4) + (1/2 x 4) + (3 x 1/5) + (1/2 x 1/5) = 12 + 2 + 3/5 + 1/10 = 14 + 6/10 + 1/10 = 14 7/10 Doing it this way (multiplying all the terms out using the distributive property of multiplication over addition) was probably more work than converting: 3 1/2 x 4 1/5 = 7/2 x (21/5) = (7 x 21)/(2 x 5) = 147/10 = 14 7/10 But there might be situations where doing the conversion is more work. For example, we could have the following situation: let there be two years where some monetary quantity increases by 2% the first year and decreases by 2% the following year. Then the total overall percentage change is (1 + 2%)(1 - 2%) = (1 + 2/100)(1 - 2/100) = 102/100)(98/100) = (102 x 98)/(100 x 100) = 9996/10000 = 0.9996 or a 0.04 % drop overall. Notice how we had to execute the multiplication 102 x 98. We could also have done the following: (1 + A)(1 - A) = 1x1 + A - A - AxA = 1 - AxA = 1 - (0.02)(0.02) = 1 - 0.0004 = 0.9996 which is a lot easier in terms of the number crunching, especially if we don't have access to a calculator. So there may be times where the conversion of a mixed number to a fraction may not be the most efficient way to proceed. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 06/04/2003 at 15:34:36 From: Natasha Subject: Mixed numbers Thank you for the response. I tried to do the same with division (using the distributive property) but it doesn't seem to be working. Is it not possible to divide mixed numbers? Date: 06/04/2003 at 18:58:16 From: Doctor Douglas Subject: Re: Mixed numbers Hi again, Natasha, There is no distributive property of division over addition, so the most you can do for a ratio of mixed numbers is A+B A B --- = --- + --- here, B and D are fractions C+D C+D C+D while A and C are whole numbers. I confess that it is more difficult to contrive an example where one would divide the mixed numbers without first converting to an improper fraction. But here's one: 12 1/2 24/2 + 1/2 25/2 ------- = ---------- = ---- = 5 2 1/2 4/2 + 1/2 5/2 12 1/2 10 + 2 1/2 ------ = ---------- = 4 + 1 = 5 2 1/2 2 1/2 Here the second method is easy because the numerator splits conveniently into two pieces, each of which is relatively easy to divide (the second term, in fact, is trivial). But you do have to recognize that 10/(2+1/2) is an easy computation: 10/2.5 = 100/25 = 4. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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