Date: 08/21/99 at 16:27:07
From: Doctor Twe
Subject: Re: Tutoring division of fractions
Hi Lee!
I came up with a few more (and I hope better) examples. The first one
is nice because it correlates directly with what we do with integers.
Here they are:
Integer example:
I went to a dairy farm and bought a 10-gallon canister of milk. The
canister won't fit in my refrigerator, so I want to pour it into
several 2-gallon jugs. How many jugs do I need?
Solution:
10
-- = 5 jugs needed.
2
Fraction example:
I went to the store and bought 1/2 gallon of milk. The container won't
fit in my refrigerator (I have a "mini-fridge"), so I want to pour it
into several 1/8-gallon (one pint) containers. How many containers do
I need?
Solution:
1/2
--- = 4 containers needed.
1/8
In both cases, we divide the total quantity of milk by the capacity of
the containers. This shows that dividing a fraction by a smaller
fraction produces a value larger than one (you need more than one of
the smaller containers).
This also demonstrates an alternative way to solve dividing fractions.
The "real world" problem can be solved using integers by converting
the quantities to pints. 1/2 gallon = 4 pints, 1/8 gallon = 1 pint.
Then 4/1 = 4 containers needed. The equivalent mathematical operation
is called "eliminating the fraction," and is accomplished by
multiplying both the dividend and divisor by a number that will
eliminate the denominators. The most efficient value to use is the
Least Common Multiple (LCM) of the denominators of the two fractions -
in this example 8.
8 1/2 8*1/2 4 pints
- * --- = ----- = ------- = 4 containers.
8 1/8 8*1/8 1 pint
A second example:
(A) How many 1/2-hour 'Simpsons' episodes can you watch in 1/2 hour?
(B) How many 1/4-hour 'Rugrats' episodes can you watch in 1/2 hour?
Solution:
(A)
1/2 1 2
--- = - * - = 1 'Simpsons' episode.
1/2 2 1
and
(B)
1/2 1 4
--- = - * - = 2 'Rugrats' episodes.
1/4 2 1
Here again, we can solve the problem by converting to a smaller unit
(minutes). 1/2 hour = 30 minutes, 1/4 hour = 15 minutes. So 30/30 = 1
'Simpsons' episode, and 30/15 = 2 'Rugrats' episodes. Eliminating the
fraction can be accomplished by multiplying both the dividend and
divisor by 30. (Note that 30 is _not_ the LCM, but it works because it
is a common multiple of 2 and 4).
(A)
60 1/2 60*1/2 30 min.
-- * --- = ------ = ------- = 1 'Simpsons' episode.
60 1/2 60*1/2 30 min.
(B)
60 1/2 60*1/2 30 min.
-- * --- = ------ = ------- = 2 'Rugrats' episodes.
60 1/4 60*1/4 15 min.
A final thought:
We eliminate the fraction in the "real world" situations by converting
to a smaller unit (pints instead of gallons, or minutes instead of
hours). We can think of the mathematical method as converting to a
smaller base unit. We are counting in 1/8ths (or /60ths, etc.) instead
of 1's.
You might also like to read the Dr. Math FAQ on dividing fractions:
http://mathforum.org//dr.math/faq/faq.divide.fractions.html
- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
Date: 08/21/99 at 16:27:07From: Doctor TweSubject: Re: Tutoring division of fractionsHi Lee!I came up with a few more (and I hope better) examples. The first one is nice because it correlates directly with what we do with integers. Here they are:Integer example:I went to a dairy farm and bought a 10-gallon canister of milk. The canister won't fit in my refrigerator, so I want to pour it into several 2-gallon jugs. How many jugs do I need?Solution: 10 -- = 5 jugs needed. 2Fraction example:I went to the store and bought 1/2 gallon of milk. The container won't fit in my refrigerator (I have a "mini-fridge"), so I want to pour it into several 1/8-gallon (one pint) containers. How many containers do I need?Solution: 1/2 --- = 4 containers needed. 1/8In both cases, we divide the total quantity of milk by the capacity of the containers. This shows that dividing a fraction by a smaller fraction produces a value larger than one (you need more than one of the smaller containers).This also demonstrates an alternative way to solve dividing fractions. The "real world" problem can be solved using integers by converting the quantities to pints. 1/2 gallon = 4 pints, 1/8 gallon = 1 pint. Then 4/1 = 4 containers needed. The equivalent mathematical operation is called "eliminating the fraction," and is accomplished by multiplying both the dividend and divisor by a number that will eliminate the denominators. The most efficient value to use is the Least Common Multiple (LCM) of the denominators of the two fractions - in this example 8. 8 1/2 8*1/2 4 pints - * --- = ----- = ------- = 4 containers. 8 1/8 8*1/8 1 pintA second example:(A) How many 1/2-hour 'Simpsons' episodes can you watch in 1/2 hour? (B) How many 1/4-hour 'Rugrats' episodes can you watch in 1/2 hour?Solution:(A) 1/2 1 2 --- = - * - = 1 'Simpsons' episode. 1/2 2 1and(B) 1/2 1 4 --- = - * - = 2 'Rugrats' episodes. 1/4 2 1Here again, we can solve the problem by converting to a smaller unit (minutes). 1/2 hour = 30 minutes, 1/4 hour = 15 minutes. So 30/30 = 1 'Simpsons' episode, and 30/15 = 2 'Rugrats' episodes. Eliminating the fraction can be accomplished by multiplying both the dividend and divisor by 30. (Note that 30 is _not_ the LCM, but it works because it is a common multiple of 2 and 4).(A) 60 1/2 60*1/2 30 min. -- * --- = ------ = ------- = 1 'Simpsons' episode. 60 1/2 60*1/2 30 min.(B) 60 1/2 60*1/2 30 min. -- * --- = ------ = ------- = 2 'Rugrats' episodes. 60 1/4 60*1/4 15 min.A final thought:We eliminate the fraction in the "real world" situations by converting to a smaller unit (pints instead of gallons, or minutes instead of hours). We can think of the mathematical method as converting to a smaller base unit. We are counting in 1/8ths (or /60ths, etc.) instead of 1's.You might also like to read the Dr. Math FAQ on dividing fractions: http://mathforum.org//dr.math/faq/faq.divide.fractions.html - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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