In division by a fraction why is it that we invert and then multiply? I posted this question so that I could explain why this works.

5 Answers
Sep 26, 2016

See the demonstration in the explanation

Solution 1 of 2
Also see my equivalent using algebra. (2 of 2)

Explanation:

#color(blue)("Preamble")#

Consider the example #1/4 -:1/8 = 2#

This is the same as:#" "(1/4color(magenta)(xx1))-:1/8=2#

#(1/4color(magenta)(xx2/2))-:1/8=2#

#(1xx2)/(4xx2)-:1/8=2#

#2/8-:1/8=2#

#color(brown)("Notice that if we just have the numerators it gives the same answer")#

#2-:1=2#

#color(brown)("However, for direct division of numerators to work the denominators must be the same")#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Using an example to demonstrate the principle")#

Selecting numbers that are obviously different.

Suppose we had: #color(brown)([2/4])color(green)(-:8/6)#

Change the 4 in #color(brown)([2/4]# such that it is in #6^("ths")" instead " -> color(red)(4)xx6/4#

For multiply or divide, what we do to the bottom we do to the top.

#color(brown)([(2xx6/4)/(4xx6/4)]) -:8/6#

Now that both the denominators are in #6^("ths")# we can ignore them and just consider the numerators.

#2xx6/4 -:8" "->" "2-:8xx6/4#

#2/8xx6/4" "->" "(2xx6)/(8xx4)#

Swap the 8 and 4 round

#(2xx6)/(4xx8)" "=" "color(brown)(2/4)color(green)(xx6/8)#

#color(purple)("The "8/6" has now been inverted and the divide ")#
#color(purple)("has become multiply. The principle has been demonstrated.")#

Sep 26, 2016

See below for an alternate (perhaps more abstract) explanation than the one provided by Tony.

Explanation:

In part this question deals with what it means to divide.

In general #color(red)(a) div color(blue)(b) = color(magenta)(k)#
means #color(red)(a)# is equivalent to #color(blue)(b)# "pieces" each of size #color(magenta)(k)#
or#color(white)("XXX")color(red)(a)=color(magenta)k xx color(blue)(b)#

When dividing by a fraction, say #color(blue)(p/q)# instead of #color(blue)(b)#
we could write #color(white)("XX")color(red)(a)divcolor(blue)(p/q)=color(magenta)(k)#
meaning #color(white)("XX")color(red)(a)=color(magenta)(k)xxcolor(blue)(p/q)#

Provided #color(green)(q/p)!=0#
we also know that we can multiply both sides of an equation by #color(green)(q/p)# and the equation will remain valid.

So
#color(white)("XXX")color(red)(a)xxcolor(green)(q/p) = color(magenta)(k)xxcolor(blue)(cancel(p)/cancel(q)) xx color(green)(cancel(q)/cancel(p)#

and since from our original specification that #color(red)(a)divcolor(blue)(p/q)=color(magenta)(k)#
it follows that
#color(white)("XXX")color(red)(a) divcolor(blue)(p/q)=color(red)(a)xxcolor(green)(q/p)#

Sep 28, 2016

#color(magenta)("I added this for completeness of my solution")#

#color(magenta)("It is a general solution")#
Solution 2 of 2
Also see my equivalent using numbers ( 1 of 2)

Explanation:

Suppose we hade #color(blue)(a/b -: c/d)#
Making the denominators so that they are all #d#

#color(blue)([a/bxx1]-:c/d)#

#color(blue)([(axx d/b)/(bxxd/b)] -:c/d)#
'...................................................................................
Note that: #" "bxxd/b" " ->" " b/bxxd" " ->" " 1xxd" "=" "d#
'......................................................................................

This becomes

#color(blue)([(axxd/b)/d] -:c/d)#

This gives the same answer as:

#color(blue)(axxd/b -:c)rarr" compare to "[4/8-:2/8]=2=[4-:2]#

#color(blue)(axxd/bxx1/c)#

#color(blue)(a/bxxd/c)#

#color(green)("Notice that the "-:c/d " has now become "xxd/c)#

#color(magenta)("Thus the rule, invert and multiply is true")#

Jan 2, 2017

An alternate but similar approach added.

Explanation:

What does a fraction when represented as #a/b# mean?
By definition it means for any two numbers #a and b#, with the condition that #b!=0#

#a# divided by #b#
or symbolically

#=>a -: b# ........(1)

We also know that #a/b# can also be written as

#a" multipled by "1/ b#
or symbolically

#=>axx1/ b# .....(2)

For expressions in lines (1) and (2) to be equal

#a -: b-=axx1/ b#

This is same as saying that

#"division is multiplication with the reciprocal"#

Jan 2, 2017

The explanation is really simple...

Consider first #24div3#

What we are actually asking is "

If I have 24 of anything, how many groups can I make with 3 in each group?"

This could be shown like this:
24 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
= (1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1 +1)+(1+1+1)

We can see that 8 possible groups can be made.

#24 div 3 = 8#

What about #4 div 1/2?#

We are asking how many groups can be made with #1/2# in each?

#4 = 1/2+1/2+1/2+1/2+1/2+1/2+1/2+1/2" "larr# there are 8

This is because each #1# has two halves in it and #4 xx2 =8#

What about #6 div 3/4?#

We are asking " how many groups of #3/4# can be made from 6"?
First we need to change #6# into quarters, and then group them into threes.

Each 1 has four quarters in it.

#6 = 6 xx4 = 24# quarters

"How many groups of 3 quarters can be made from 24 quarters?"

#6 = (1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)#

There are 8 groups with three quarters in each.

#6 div 3/4 = 8#

What did we do? We changed everything into quarters by multiplying by 4 and then divided by 3 to make groups of #3/4#

#6 div 3/4 = 6 xx4 div 3#

# = 6 xx4/3#

#=8#

In the same way. #4 div 2/5#

Make everything into fifths by #xx5#, then #div2# to make groups of #2/5#

#4 div 2/5 = 4 xx5 div 2#

#=4 xx5/2#

#=20/2#

#=10#