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Invert and Multiply

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Date: 11/08/97 at 14:24:45
From: Anonymous
Subject: Fractions!

Hi Doc,

I am a student teacher, currently taking a methods course in
elementary mathematics. I am struggling with how to explain to a class
"why" we invert and multiply when dividing fractions. I read your FAQ
on this, but still don't understand how I would explain it using

Thanks ever so much!
Laura
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Date: 11/09/97 at 09:56:11
From: Doctor Chita
Subject: Re: Fractions!

Hi there:

Explaining "why" using concrete materials may be difficult. However,
this is an opportunity, perhaps, to show students how mathematics
relies on proof, rather than pictures or models.

Take an "easy" problem such as, 1/2 divided by 3/4.

Write this as a compound fraction: (1/2)/(3/4). (Use horizontal
fraction bars, not diagonal bars as here. The computer has its
handicaps!)

Everyone knows that 1 is the multiplicative identity element.
Therefore,

1 * (1/2)/(3/4) = (1/2)/(3/4)

Substitute 4/4 for 1. Write:

4/4 * (1/2)/(3/4)

Think of 4/4 as the compound fraction: (4/1)/(4/1). Then multiply the
numerator and deminator of the compound fraction by 4/1:

[4/1 * (1/2)]/4/1* (3/4)]

The new numerator is 2: the new denominator is 3. The answer is 2/3.

Therefore, the rule "invert and multiply" works because what you are
really doing is multiplying the numerator and denominator of a
compound fraction by the LCD of the fractions in the compound
fraction.

Try 3/8 divided by 9/5. Write as (3/8)/(9/5) using horizontal bars.

Multiply the fraction by 1 = 40/40, in the form of (40/1)/(40/1) where
40 is the LCD of 8 and 5:

(40/1)/(40/1) * (3/8)/(9/5)

[40/1 * 3/8]/ [40/1 * 9/5]

15/72 = 5/24

After a while, you can drop the 1s in the fraction that represents the
LCD.

Got it? Put this information together with what's in the archives and
write back if you still need more help. Good luck.

-Doctor Chita,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Elementary Fractions
Middle School Fractions

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