Question

How do you divide `\frac { \frac { 5} { 12} + \frac { 1} { 3} } { \frac { 7} { 12} - \frac { 1} { 2} }`?

Answer 1

9

Explanation:

First, let's sum the fractions

`5/12+1/3` and `7/12-1/2` and get:

`((5+4)/cancel12)/((7-6)/cancel12)`

and simplify:

`9/1=9`

Answer 2

`9`

A lot of detail given using first principles showing the methods that are behind the shortcuts approach.

Explanation:

`color(blue)("Consider "5/12+1/3)`

`color(green)(5/12+[color(white)(.)1/3color(red)(xx1)" "]`

`color(green)(5/12+[color(white)(.)1/3color(red)(xx4/4)" "]`

`5/12+4/12=9/12`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider "7/12-1/2)`

`color(green)(7/12-[color(white)(.)1/2color(red)(xx1)" "])`

`color(green)(7/12-[color(white)(.)1/2color(red)(xx6/6)" "])`

`7/12-6/12 = 1/12`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together")`

`(color(white)(..)9/12color(white)(..))/(1/12)`

`9/12-:1/12`

You would have been shown that this is the same as:

`color(red)(9/12)xxcolor(green)(12/1)`

`(color(red)(9)xxcolor(green)(12))/(color(red)(12)xxcolor(green)(1))`

`(color(red)(9)xxcolor(green)(12))/(color(green)(1)xxcolor(red)(12))`

`9/1xx12/12`

But `12/12=1 larr" This is what cancelling is all about"`

`9/1xx1=9`