Monday, September 7, 2015

Dividing Fractions

Few people remember how to divide fractions, and even less remember the reasoning behind it.  In fact, there are many math teachers who do not even know the reasoning behind the method for dividing fractions.  Hence the saying, “Ours is not to reason why; just invert and multiply.”  Most students accept the rule of inverting and multiplying to divide a fraction as a magical shortcut that miraculously works, use it on their worksheet and test, and then promptly forget it.  However, a proper understanding behind the method will result in a better memory of the method and a better understanding of the subject.  Therefore, we will reason why we invert and multiply.



Changing Units Proof

Let’s start with a simple example: “How many quarts are in two gallons?”  Since 1 quart is a 1/4 of a gallon, this is equivalent to asking what 2 gallons divided by 1/4 of a gallon is, or 2 ÷ 1/4.  However, it is much easier to solve the question by using quarts instead of gallons.  Since there are 4 quarts in a gallon, 2 gallons is 8 quarts, therefore, we can ask the equivalent question, “How many quarts are in 8 quarts?”, or 8 ÷ 1, which is much easier to solve.


Here’s another example: “Your favorite TV game show, Math Mania, has episodes that are 2/3 of an hour long.  How many episodes of Math Mania can you watch in 2 hours?”  Of course, you could set up the problem as 2 hours divided by 2/3 of an hour, or 2 ÷ 2/3, but it is much easier to set up the problem as 120 minutes divided by 40 minutes, or 120 ÷ 40 = 3.  In fact, it would be even better if we made up a unit of time that is worth 20 minutes which for lack of a better word we will call a “zoikle”.  A zoikle is then 1/3 of an hour, so 2 hours is 6 zoikles, and each episode of Math Mania that is 2/3 of an hour long is 2 zoikles long.  Therefore, we can set up the problem as 6 zoikles divided by 2 zoikles, or 6 ÷ 2 = 3.


7:00 pm
7:30 pm
8:00 pm
8:30 pm
314
MBC
Math Mania
(2/3 hr)
Math Mania
(2/3 hr)
Math Mania
(2/3 hr)
315
XYZ
The Fraction Family (2/5 hr)
The Fraction Family (2/5 hr)
The Fraction Family (2/5 hr)
The Fraction Family (2/5 hr)
The Fraction Family (2/5 hr)
317
MNC
The Walking Divisors
(1 hr)
Guy’s Graphing Games
(1 hr)

In both cases, we changed the units to something more convenient.  In the quart-gallon problem, we changed gallons to quarts, so 2 ÷ 1/4 became 8 ÷ 1 because 2 ÷ 1/4 = (2 · 4) ÷ (1/4 · 4) = 2 · 4 ÷ 1 = 2 · 4/1, which is inverting and multiplying.  In the game show problem, we changed hours to zoikles (20 minutes), so 2 ÷ 2/3 became 6 ÷ 2 because 2 ÷ 2/3 = (2 · 3) ÷ (2/3 · 3) = 2 · 3 ÷ 2 = 2 · 3/2, which is also inverting and multiplying. 

Here’s a tougher example that does not work as cleanly as the other two: “Your favorite TV sitcom, The Fraction Family, has episodes that are 2/5 of an hour long.  How many episodes of The Fraction Family can you watch in 13/4 hours?”  Of course, you could set up the problem as 7/4 hours divided by 2/5 of an hour, or 7/4 ÷ 2/5.  This time, however, converting to minutes is not easier, because it results in the two-digit long division problem of 105 minutes divided by 24 minutes, or 105 ÷ 24.  So we will use a unit of time that is the common denominator of 7/4 and 2/5, namely 1/20 of an hour, which we will now call a “zazzer”.  Since a zazzer is 1/20 of an hour, 7/4 hours is 7/4 · 20 or 35 zazzers, and each episode of The Fraction Family that is 2/5 of an hour long is 2/5 · 20 or 8 zazzers long.  Therefore, we can set up the problem as 35 zazzers divided by 8 zazzers, or 35 ÷ 8 = 4 3/8.  By changing hours to zazzers, 7/4 ÷ 2/5 became 35 ÷ 8 because 7/4 ÷ 2/5 = (7/4 · 20) ÷ (2/5 · 20) = (7 · 1/4 · 20) ÷ (2 · 1/5 · 20) = (7 · 5) ÷ (2 · 4) = 7·5/2·4 = 7·5/4·2 = 7/4 · 5/2, which is inverting and multiplying.

This method can be applied as a general rule for any fraction by converting to a new unit equal to the common denominator of the two divided fractions.  In other words, a/b ÷ c/d becomes a/b · d/c because a/b ÷ c/d = (a/b · bd) ÷ (c/d · bd) = (a · 1/b · bd) ÷ (c · 1/d · bd) = (a · d) ÷ (c · b) = a·d/c·b = a·d/b·c = a/b · d/c, which is inverting and multiplying.


Fractions Within a Fraction Proof

Another way to prove the invert and multiply method is to rewrite the problem as fractions within a fraction.  For example, to solve 7/4 ÷ 2/5, we can rewrite the division symbol as a fraction symbol, or 7/4 / 2/5.  Now we want to change the denominator 2/5 to 1, and to do that we need to multiply the numerator fraction and denominator fraction by the reciprocal of 2/5, or 5/2.  So:





7

7
·
5

7
·
5




7
÷
2
=
4
=
4

2
=
4

2
=
7
·
5
4

5

2

2
·
5


1


4

2




5

5

2









which is inverting and multiplying.

The general case follows the same logical steps:





a

a
·
d

a
·
d




a
÷
c
=
b
=
b

c
=
b

c
=
a
·
d
b

d

c

c
·
d


1


b

c




d

d

c









which is also inverting and multiplying.


Algebraic Proof

A third way to prove the invert and multiply method is to use algebra.  This proof is probably the easiest to show to adults, but unfortunately the rules of algebra are typically taught to children several years after the rules of fractions are taught, so this proof is not helpful for most elementary students.

To solve 7/4 ÷ 2/5 using the rules of algebra, first let x = 7/4 ÷ 2/5.  Multiplying 2/5 on both sides, the equation becomes 2/5x = 7/4.  Multiplying both sides by 5 and dividing both sides by 2 gives us x = 7/4 · 5/2, which is inverting and multiplying.  Again, the general case follows the same logical steps. To solve a/b ÷ c/d, first let x = a/b ÷ c/d.  Multiplying c/d on both sides, the equation becomes c/dx = a/b.  Multiplying both sides by d and dividing both sides by c gives us x = a/b · d/c, which is also inverting and multiplying.


Conclusion

It is never a good idea to just accept a statement without explanation, not only in mathematics but also in life in general, even if the statement seemingly works all the time.  This also applies to the rule for dividing fractions by inverting and multiplying.  We ought to reason why we invert and multiply and not just accept it as a magical method.  In so doing, hopefully it will stay in our memory longer and give us a deeper understanding of fractions.

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