How do you divide `\frac { 3} { 4} \div \frac { 5} { 4}`?

Dividing is the same as multiplying by the inverse.

So
`color(white)("XXX")3/4div5/4`
is the same as
`color(white)("XXX")3/4xx4/5`

`color(white)("XXX")=3/5`

The complex fraction method makes sense of the rule of invert and multiply.

`3/4-:5/4` can be written as

`(3/4) / ( 5/4)` This is a complex fraction

To simplify the fraction multiply both the numerator ( top fraction) and the denominator ( bottom fraction) by the inverse of the denominator .

The inverse of `5/4 = 4/5` Now multiply both fractions by `4/5`

`{ ( 3/4) xx (4/5)} / {(5/4) xx ( 4/5)}`

`( 5/4) xx ( 4/5) = 1` This causes the bottom fraction to disappear. This leaves only the top fraction which gives.

`( 3/4) xx ( 4/5) = ( 12/20)` Now factor out the common factor of 4

`( 4 xx 3)/ (4 xx 5)` This leaves

`3/5`

After the bottom fraction divides out it leaves the invert and multiply method. Using the complex fraction makes mathematical sense of the method and avoids reliance on memory.

A fraction consists of `("count")/("size indicators of what you are counting")`

Using the allocated names we have:

`("count")/("size indicator")->("numerator")/("denominator")`

Consider whole numbers. For example 6 and 3

These can, and may, be written as `6/1 and 3/1` They are rational numbers. It is not normally done but never the less it is correct.

Now consider `6-:3->6/1-:3/1`

`color(blue)("You just divide the counts as the 'size indicators' are the same")`

`6-:3->6/1-:3/1->6/3=2`

`color(magenta)("You have been applying this principle for a long time without even realizing that you were.")`