Question

How do you write `10` `5/12` as an equivalent improper fraction?

Answer 1

`color(brown)("This has been shown using first principles. The ")` `color(brown)("shortcut methods do not explain what is happening.")`

`" "color(magenta)(10 5/12 = 125/12)`

Explanation:

Given:`" "10 5/12`

`color(blue)("Shortcut method - with proper sign")`

`10 5/12 = (10xx12)/12+5/12color(blue)( =125/12)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("First principles with full explanation")`

This can be written as:`" " 10+5/12`

Although not normally done, it is perfectly correct to write `10" "as" " 10/1`

So we now have:`" " 10/1+5/12`

To be able to add the top numbers (numerator) directly the bottom numbers (denominators) need to be the same.

If we multiply a number by 1 then its value is not changed and it still looks the same.

However, the value of 1 can be presented in many ways. For example `1=12/12`

Consider the `10/1` Let us multiply it by `1=12/12` so we have

`" "10/1xx12/12=(10xx12)/(1xx12) = 120/12`

The `120/12` is exactly the same value as `10/1` but it just looks different.

So now we can write

`" "10 5/12 = 10/1+5/12 = color(brown)(120/12+5/12)`

`" "color(magenta)("Now you can directly add the top numbers.")`

`" "120/12+5/12 =(120+5)/12 = 125/12`

`" "color(magenta)(10 5/12 = 125/12)`