Question

If `13m=5n`, then what is `m/n`?

Answer 1

`m/n=5/13`

Explanation:

Divide both sides by `n`.

`(13m)/n=(5n)/n`

The `n` terms will cancel on the right hand side.

`(13m)/n=(5color(red)(cancel(color(black)(n))))/color(red)(cancel(color(black)(n)))`

`(13m)/n=5`

Divide both sides by `13`. Recall that this is the same as multiplying both sides by `1//13`.

`1/13((13m)/n)=5(1/13)`

Now, on the left side, we see that the `13`s will cancel, since they are present in the numerator and the denominator.

`1/color(red)(cancel(color(black)(13)))((color(red)(cancel(color(black)(13)))m)/n)=5/13`

`m/n=5/13`

Answer 2

`" "m/n=5/13`

Explanation:

Write as `13xxm=5xxn`

`color(blue)("Shortcut method")`

Using the principle that if you move something that multiplies to the other side of the = it becomes divide.

By sight: `color(green)(m/n=5/13)`

`color(brown)("Not that the shortcut method comes from first principles")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Using first principles")`

Divide both sides by n giving

`" "13xxm/n=5xx n/n`

But `n/n=1` giving

`" "13xxm/n=5`

Divide both sides by 13 giving

`" "13/13xxm/n=5/13`

But `13/13 = 1` giving

`" "color(green)(m/n=5/13)`