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### Multiplying Mixed Numbers

```Date: 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/
```
Associated Topics:
Elementary Division
Elementary Fractions
Elementary Multiplication

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