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.
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7:00 pm
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7:30 pm
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8:00 pm
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8:30 pm
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314
MBC
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Math Mania
(2/3
hr)
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Math Mania
(2/3
hr)
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Math Mania
(2/3
hr)
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315
XYZ
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The Fraction
Family (2/5 hr)
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The Fraction
Family (2/5 hr)
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The Fraction
Family (2/5 hr)
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The Fraction
Family (2/5 hr)
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The Fraction
Family (2/5 hr)
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317
MNC
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The Walking
Divisors
(1 hr)
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Guy’s Graphing
Games
(1 hr)
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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:
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7
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7
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5
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7
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5
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7
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÷
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2
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4
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4
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4
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5
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5
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5
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1
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4
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5
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5
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2
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which is
inverting and multiplying.
The general
case follows the same logical steps:
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a
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a
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d
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a
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d
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a
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÷
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c
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=
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b
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=
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b
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c
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=
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b
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c
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=
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a
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d
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b
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d
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c
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c
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1
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b
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c
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d
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d
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c
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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.
